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\sbasedon0\snext13 refs;}{\s14\qj\sl-240 \f20 \sbasedon0\snext14 fixed 12;}}{\info}\paperw11880\paperh16820\margl1701\margr1701\margt1447\margb1447\facingp\deftab709\widowctrl\ftnbj\ftnrestart \sectd \linemod0\linex0\cols1\endnhere \pard\plain
\s255\qj\ri29\sb240\sl280\tx720\tqc\tx3960 \b\f20\fs28 \tab \tab PROGRAM \ldblquote UNIMOL\rdblquote \par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard\plain \s254\qc\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 CALCULATION OF RATE COEFFICIENTS FOR UNIMOLECULAR AND RECOMBINATION REACTIONS\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qc\ri29\sl280\tx720\tqc\tx3960 Authors: Robert G Gilbert, Meredith J T Jordan and Sean C Smith\par
School of Chemistry, Sydney University, NSW 2006. Australia\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qc\ri29\sl280\tx720\tqc\tx3960 {\f23 \rdblquote }Robert G Gilbert, Meredith J T Jordan, Sean C Smith, Ian G Pitt, Paul G Greenhill 1992\par
version of June 1993\par
\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 Referencing: this program should be referenced as:\par
\pard \qc\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\li720\ri29\sl280\tx720\tqc\tx3960 Gilbert, R. G.; Smith, S. C.; Jordan, M. J. T., {\i UNIMOL program suite (calculation of fall-off curves for unimolecular and recombination reactions)}
(1993). Available from the authors: School of Chemistry, Sydney University, NSW 2006, Australia or by email to: gilbert_r@summer.chem.su.oz.au.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 Enquiries and requests for upgrades (produced approximately once per year) should be addressed to R G Gilbert, preferably by email to: {\fs28 gilbert_r@summer.chem.su.oz.au}.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b \tab \tab Contents\par
}\pard \qc\ri29\sl280\tx720\tqc\tx3960 {\b \par
}\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard\plain \s233\qj\fi630\li90\ri-3\sl280\tqr\tldot\tx6480 \f20\fs18 BASIC THEORY\tab 1\par
PROGRAM STRUCTURE\tab 5\par
PROGRAM RRKM\tab 6\par
PROGRAM MASTER\tab 11\par
PROGRAM GEOM (co-authorsI G Pitt, P G Greenhill)\tab 14\par
PROGRAM BRW\tab 16\par
APPENDIX I (high-pressure limit - simple fission)\tab 18\par
APPENDIX II (dihedral angles)\tab 18\par
NOTES\tab 25\par
\pard \s233\qj\fi630\li90\ri-3\sl280\tqr\tldot\tx6480 Index\tab 31\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\b \sect }\sectd \pgnrestart\linemod0\linex0\cols1\colsx0 {\headerl \pard\plain \s244\qj\ri29\sl280\tqc\tx3969\tqr\tx8504 \f20\fs20 {\chpgn }\tab {\i\fs18 UNIMOL program description}\par
}{\headerr \pard\plain \s244\qj\ri29\sl280\tqc\tx3969\tqr\tx7920\tqr\tx8504 \f20\fs20 \tab {\i\fs18 UNIMOL program description\tab }{\chpgn }\par
}\pard\plain \s254\qc\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 OVERVIEW\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 This FORTRAN program package enables one to calculate the pressure and temperature dependence of unimolecular and {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20
recombination (association) rate coefficient}}s. It employs RRKM{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v RRKM theory}}} theory and a numerical solution of the master equation (\ldblquote weak collision\rdblquote
falloff effects). The programs are called RRKM and MASTER (which uses a file prepared by RRKM). Additional programs (GEOM and BRW) are also supplied to assist in the calculation of
RRKM and collisional energy transfer parameters. Calculations can be performed for a single reaction (A {\f23 \'ae} B) and also for multi-channel reactions (A {\f23 \'ae} B, A {\f23 \'ae}
C). The package can be used for tight and loose transition states, including radical-radical{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v recombination (association) rate coefficient}}} and ion-molecule associations.{\v {\xe
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v ion-molecule reaction}}}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\li680\ri29\sl280\tx720\tqc\tx3960 {\b The basic theory used by this program is explained in the text \ldblquote Theory of Unimolecular and Recombination Reactions\rdblquote
, by R G Gilbert and S C Smith, Blackwell Scientific Publications, Oxford (U.K.) and Cambridge (Mass.), 1990.}
The following gives a very brief summary; the text should be consulted before use of this program. This program updates, and completely replaces, the original FALLO
FF program 460, QCPE 3, 64 (1983), and subsequent modifications by Shandross and Howard in QCPE 530 and by Gilbert and Smith. This program is copyright; free updates can be obtained by users who have email, by sending a request to GILBERT_R@SUMMER.CHEM.SU.
OZ.AU{\v \par
}\pard \qj\ri29\sl280\tx720\tqc\tx3960 It would be especially appreciated if the authors can be notified by users of bugs and problems with the program, and of suggestions for improvements to this program description.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 Using this program, one can predict or fit rate data at any temperatur
e and pressure for unimolecular and recombination (association) reactions. One can for example (i) fit experimental falloff data obtained over a limited pressure range to estimate high-pressure parameters; (ii) use established high-pressure parameters to c
alculate the pressure dependence of the rate coefficient; (iii) use tabulated RRKM and energy transfer parameters from the literature to calculate rate coefficients at any pressure and temperature; (iv) fit falloff and high-pressure rate data to obtain RRK
M and energy transfer parameters; (v) for a new reaction, carry out an {\i a priori}
estimate of RRKM and energy transfer parameters so as to predict the rate coefficient under desired conditions of pressure and temperature; (vi) test experimental data by fitting with these programs and then examining the resulting RRKM and energy transfe
r parameters to see if they lie within physically acceptable ranges for the type of system under study.\-\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 The program suite has been specifically written for a Vax (using the G_FLOATING option in the {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 Fortran compiler}}
). However, no difficulty has been experienced in transfer to other machines, except that there may be changes required in FORMAT, PROGRAM, OPEN and CLOSE statements depending on the particular machine. The code is in double precision (although on machines
with large word length, only single precision will be necessary). Now, {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 recombination (association) {\v rate coefficient}}}
rate coefficients are calculated from the equivalent unimolecular rate coefficients using mi
croscopic reversibility; since unimolecular rate coefficients for typical recombination (association) reactions are usually very small at room temperatures, it is necessary when calculating recombination rate coefficients under such conditions to be able t
o work with very small floating point numbers (say, 1D-60); this may pose a problem with some Fortran compilers. The program suite runs successfully on a PC with Microsoft Fortran (Version 4.0 or above).\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 {\v {\tc\tcl1\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 \tab \tab BASIC THEORY}}\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 This program package enables on
e to calculate the rate coefficient for a unimolecular isomerization or decomposition. It can also be used to calculate a recombination (association) rate coefficient. A thermal unimolecular decomposition (AB {\f23 \'ae} A + B) and the corresponding {\v
{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 recombination (association){\v rate coefficient}}} (A + B {\f23 \'ae}
AB) rate process are the reverse of each other, and therefore the respective rate coefficients are exactly related from statistical thermodynamics as\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab \|F({\i k}{\fs18\dn4 rec},{\i k}{\fs18\dn4 uni}) = {\i K}{\fs18\dn4 eq} = 5.322{\f23 \'b4}10{\fs18\up6 \endash 21 }\|b(\|f({\i m}{\fs14\dn4 A}+{\i m}{\fs14\dn4 B},{
\i Tm}{\fs14\dn4 A}{\i m}{\fs14\dn4 B})){\fs16\up10 \|S\|UP8(3)\|S\|UP7(/)\|S\|UP6(2)}\|f({\i Q}{\fs14\dn4 AB},{\i Q}{\fs14\dn4 A}{\i Q}{\fs14\dn4 B}) e\|s\|up8({\f23 D}{\i H}{\fs14\dn4 0}{\f23 \'b0}/{\i k}{\fs18\dn4 B}{\i T})\tab (1)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 In the second part of eq 1, {\f23 D}{\i H}{\fs14\dn4 0}{\f23 \'b0} is the enthalpy difference between reactant and product at {\i T} = 0 K, {\i K}{\fs18\dn4 eq} has units of cm{\fs18\up6 3}
and the masses {\i m}{\fs18\dn4 A} and {\i m}{\fs18\dn4 B} are in a.m.u., and the partition functions {\i Q}{\fs18\dn4 A}, {\i Q}{\fs18\dn4 B} and {\i Q}{\fs18\dn4 AB}
include the vibrational, rotational and electronic factors only. Thus we can always find a {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 recombination (association) rate coefficient}}
given the unimolecular rate coefficient for the reverse reaction, using the value of {\i k}{\fs18\dn4 uni} produced by the program.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard\plain \s4\qj\li1080\ri360\sl280\sbys\tqc\tx3960\tqr\tx7920 \f20\fs20 energy \par
{{\pict\macpict\picw255\pich166\picscaled
0381ffffffff00d201421101a0008201000affffffff00d201422200d0008c48002200b5008c4800220091008c480022007f008c480022000a008c480022002e008c4800220040008c4800220064008c480022005b008c4800220049008c4800220013008c4800220013008c4800220013008c4800220037008c4800220052
008c480022001c008c4800220025008c480022001c009509002200130095090022000a008c1200220001008c4800a100d80004ffff8000a100d60004ffff8000a000d7a100b40005000002090908001720005b0068005b00f8a000b5a100d60004ffff8000a000d7a000d9a100d60004ffff8000a100d80004ffff8000a000
bea000d9a0008c08000822005b0068080022005b007a080022005b008c080022005b009e080022005b00b0080022005b00c2080022005b00d4080022005b00e60800a0008da100d80004ffff8000a000bfa000d9a000d7a0008c71001e005b00fe00670104005b01010067010400670101006700fe005b010171001e00c400
fe00d0010400d0010100c400fe00c4010100c4010400d001012200670101005da0008da0008c71001e000a006e00160074000a007100160074001600710016006e000a007122001600710018a0008da0008c71001e00460110004c011c0049011c004c011000490110004601100049011ca100d80004ffff8000a100d60004
ffff8000a000d7a100b40005000002090908001722004900dd3300a000b5a100d60004ffff8000a000d7a000d9a100d60004ffff8000a100d80004ffff8000a000bea000d908000822004900dd080022004900ef080022004901010800a100d80004ffff8000a000bfa000d9a000d7a0008da0008c71001e00070134000d01
40000a0140000d0134000a013400070134000a0140a100d80004ffff8000a100d60004ffff8000a000d7a100b40005000002090908001722000a00dd5700a000b5a100d60004ffff8000a000d7a000d9a100d60004ffff8000a100d80004ffff8000a000bea000d908000822000a00dd080022000a00ef080022000a010108
0022000a0113080022000a01250800a100d80004ffff8000a000bfa000d9a000d7a0008da0008c71001e00220077002e007d002e007a002200770022007a0022007d002e007a22000a007a0018a0008da0008c71001e0001000100120009000100050012000900120005001200010001000507000200022000cf0004001100
04a0008da00083ff}} \par
\pard \s4\qj\li1080\ri360\sl280\sbys\tqc\tx3960\tqr\tx7920 \par
\pard \s4\qj\li1350\ri360\sl280\sbys\tx720\tqc\tx3960\tqr\tx7920 {\i E}{\i\fs18\dn4 \par
}\par
{\i R}({\i E,E'}) {\i k}({\i E}) (reaction)\par
(collision) \|S\|DO10({\i E'})\par
\par
\par
\par
\par
{\i E}{\fs14\dn4 0}\par
\pard \s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Fig. 1.} The {\v {\xe\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 Lindemann-Hinshelwood mechanism}}
showing collision and reaction events among levels of reactant A with energy {\i E}.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 A thermal unimolecular reaction involves the following processes (Fig. 1): collisional excitation of a reactant molecule with internal energy {\i E'} to one with internal energy {\i E}
; collisional relaxation (the reverse of the previous process), and, if the internal energy is sufficiently high, conversion to product. The value of {\i k}{\fs18\dn4 uni}
at any pressure can be calculated exactly from the solution to the integral eigenvalue {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 master equation}}\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab \endash {\i k}{\fs18\dn4 uni} {\i g}({\i E}) = [M] \|O(\|S\|DO12(\|D\|BA3(){\fs18 0}),\|S\|UP16(\|D\|FO2(){\fs18 \'b0}),{\fs36\dn4 \'ba})\|B\|BC\|[({\i R}({\i E,E'})
{\i g}({\i E'})\|D\|FO1()\endash \|D\|FO1(){\i R}({\i E',E}){\i g}({\i E})) d{\i E'} \endash {\i k}({\i E}){\i g}({\i E})\tab (2)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 where {\i k}({\i E}) is the microscopic reaction rate coefficient, {\i R}({\i E,E'}\|D\|FO1()) is the rate coefficient for collisional energy transfer from energy {\i E'} to energy {\i E}
, [M] is the concentration of bath gas, and the eigenfunction {\i g}({\i E}) is the population of molecules with energy {\i E}. Alternatively, one may write {\i R}({\i E,E'}\|D\|FO1()) in terms of the {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960
\f20\fs20 probability of energy transferred per collision}}, {\i P}({\i E,E'}\|D\|FO1()), defined by:\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab {\i P}({\i E,E'}\|D\|FO1()) = \|F([M]{\i R}({\i E,E'}\|D\|FO1()),{\i\f23 w}({\i E'}\|D\|FO1()))\tab (3)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 where {\i\f23 w} is a reference collision frequency (e.g., gas-kinetic hard sphere or Lennard-Jones). Eq 3 implies the normalization condition:\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab \|O(\|S\|DO12(\|D\|BA3(){\fs18 0}),\|S\|UP16(\|D\|FO2(){\fs18 \'b0}),{\fs36\dn4 \'ba}){\i P}({\i E,E'}\|D\|FO1()) d{\i E} = 1\tab (4)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 Eq 2 can be re-written as:\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab \endash {\i k}{\fs18\dn4 uni }{\i g}({\i E}) = {\i\f23 w }\|O(\|S\|DO12(\|D\|BA3(){\fs18 0}),\|S\|UP16(\|D\|FO2(){\fs18 \'b0}),{\fs36\dn4 \'ba})\|B\|BC\|[({\i P}({
\i E,E'}\|D\|FO1()){\i g}({\i E'}\|D\|FO1()) \endash {\i P}({\i E',E}){\i g}({\i E})) d{\i E'} \endash {\i k}({\i E}){\i g(E})\tab (5)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 The value of {\i k}{\fs18\dn4 uni} at any pressure can be found either directly from eqs 2 or 5 or from:\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab {\i k}{\fs18\dn4 uni} = \|F(\|O(\|S\|DO12(\|D\|BA3(){\i\fs18 E}{\fs14\dn4 0}),\|S\|UP16(\|D\|FO2(){\fs18 \'b0}),{\fs36\dn4 \'ba}){\i k}({\i E}){\i g}({\i E}) d{\i E}
,\|O(\|S\|DO12(\|D\|BA3(){\fs18 0}),\|S\|UP16(\|D\|FO2(){\fs18 \'b0}),{\fs36\dn4 \'ba}){\i g}({\i E}) d{\i E})\tab (6)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
Here {\i E}{\fs14\dn4 0} is the critical energy for reaction, and {\i g}({\i E}) is found from the solution of eqs 2 or 5.\par
\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 THE \ldblquote FALLOFF\rdblquote PROGRAM PACKAGE COMPUTES {\i k}({\i E}) FROM RRKM THEORY (program \ldblquote RRKM{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v RRKM theory}}}\rdblquote
), SOLVES EQ 3 TO GIVE {\i k}{\fs18\dn4 uni} (program \ldblquote MASTER\rdblquote ) AND GENERATES THERMODYNAMIC DATA FOR EQ 1 (program \ldblquote RRKM\rdblquote ) TO ENABLE {\i k}{\fs18\dn4 rec} TO BE CALCULATED. A PROGRAM \ldblquote GEOM\rdblquote
IS ALSO SUPPLIED TO CALCULATE DATA ON ROTATIONAL CONSTANTS, ESPECIALLY THOSE INVOLVING HINDERED ROTORS, AS REQUIRED IN RRKM CALCULATIONS. PROGRAM \ldblquote BRW\rdblquote GIVES A SIMPLE MODEL (THE BIASED RANDOM WALK APPROACH) WHICH ENABLES {\i P}({\i E}
,{\i E'}\|D\|FO1()) TO BE ESTIMATED FROM MOLECULAR PROPERTIES.{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v biased random walk model}}}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 Strictly speaking, the above equations should not only be written down in terms of energy{\i E} but also in terms of angular momentum {\i J}
, both of which are conserved in an isolated molecule. For simplicity the complications due to {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v angular momentum conservation}}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v
J-conservation}}}conservation of angular momentum are not discussed in detail in this brief introduction, but are however incorporated into the program suite.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard\plain \s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 {\b The reaction step} RRKM{\v {\xe\pard\plain \s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 {\v RRKM theory}}} theory (i.e., microcanonical {\v {\xe\pard\plain
\s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 transition state theory}}) is used to generate the {\i k}({\i E}), from the expression:{\v {\xe\pard\plain \s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 {\v microscopic reaction rate}}}
\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 {\i \tab \tab k}({\i E}) = \|F(\|I({\fs18\dn4 0}{\fs18 ,}\|S\|DO2({\i\fs18 E}{\fs18 \endash }{\i\fs18 E}{\fs14\dn4 0}), {\i\f23 r}{\fs14\up6 \'a0}({\i E}{\fs14\dn4 +}) d{\i E}
{\fs14\dn4 +}), {\i h} {\i\f23 r}({\i E})) \tab (7)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 Here {\i h} is Planck\rquote s constant, and {\i\f23 r}({\i E}) and {\i\f23 r}{\fs18\up6 \'a0}(E) are the densities of states of the reactant molecule and of the activated complex (\ldblquote activated complex
\rdblquote is synonymous with \ldblquote transition state\rdblquote ). The densities of states in turn require the vibrational frequencies and rotational constants of the species. {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 RRKM theory}
} thus requires (i) {\i E}{\fs14\dn4 0}, the critical energy for reaction, (ii) the vibrational frequencies of reactant, (iii) the vibrational frequencies of activated complex, (iv) the moments of inertia of reactant and (v)\~
the moments of inertia of product (both for external rotations and any active rotors \endash internal or external). In addition, if one uses the full angular momentum conservation{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v
angular momentum conservation}}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v J-conservation}}} treatment (applicable to {\v i.}simple-fission transition state{\v ;}s), some further information about the potential needs to be provided.\-
\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 The total number of degrees of freedom of the activated complex(es) must be one less than that of the reactant molecule; the program checks for this, and aborts with a warning if this condition is not satisfied.\-
Means of finding the requisite data are given in Gilbert and Smith. M{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v m}oments of inertia}}
of reactant and of activated complex (including any hindrance effects) can be obtained from reactant or transition state geometry using GEOM
. Note that some care is required if (as is physically realistic) it is assumed that one external rotational degree of freedom can exchange energy with vibrations (i.e., is an active rotor); this point is discussed in Note C. Because any estimates of the
frequencies and rotational constants whose values are unknown should be checked against any available thermodynamic data (entropy, etc.), calculated thermodynamic quantities are printed by the RRKM program. The RRKM calculation is carried out using a conve
ntional direct count method extended to handle rotors semi-classically.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 If the reaction involves a {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 simple-fission transition state}}: i.e., one where there is no barrier to {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960
\f20\fs20 recombination {\v (association) rate coefficient}}} (e.g., decomposition to two free radicals, or radical/radical recombination, or an {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 ion-molecule association}}
) then the rate coefficient is very sensitive to the position of the transition state. It is recommended that the position of the transition state be chosen using canonical variationa\'1btransition state theory.{\v {\xe\pard\plain
\qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v variational selection of transition state}}} This is because transition state (RRKM{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v RRKM theory}}}
) theory is an exact upper bound to the rate coefficient, and hence the transition state can be best chosen (in the absence of a barrier to {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 recombination {\v (association) rate coefficient}}}
) as the distance between moieties which minimizes the calculated rate coefficient. This can be carried out easily using GEOM: only a single input quantity (the length of the breaking bond) is
changed, and all moments of inertia, hindrances, etc., are calculated automatically, thereby rendering it simple to change the input to RRKM. See also Appendix I; note that {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 J-conservation}{\xe
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v angular momentum conservation}}} is particularly important for this class of transition states.\-\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b The collision step} Although no generally accepted theory for collisional energy transfer [analogous to RRKM theory for {\i k}({\i E}
)] has yet emerged, general principles have evolved which enable one to estimate the appropriate energy transfer parameters so as to be able to compute {\i k}{\fs18\dn4 uni} and {\i k}{\fs18\dn4 rec} to acceptable accuracy.\- Until the 1970\rquote
s, it was conventional to use the strong collision form for {\i R}({\i E,E'}\|D\|FO1()). However, it is now realized that this is physically incorrect and (even when partially repaired by allowing for a collision efficiency {\i\f23 b}
) can lead to large errors. The strong collision assumption was made historically because of the apparent difficulty of solving eq 2; new methods, such as that given in the present program, eli
minate this problem, and one should now always use a physically reasonable form for {\i R}({\i E,E'}\|D\|FO1()). Such a form is the \ldblquote weak collision\rdblquote model, which is very general: assuming that {\i P}({\i E,E'}\|D\|
FO1()) depends on both {\i E} and {\i E'} and approaches zero for sufficiently large |{\i E} \endash {\i E'}\|D\|FO1()|.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 It turns out that {\i k}{\fs18\dn4 uni} is not sensitive to the details of {\i P}({\i E,E'}\|D\|FO1()), but only to a single moment, such as the mean energy transferred per collision, <{\f23 D}{\i E}>:{\v {\xe
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v mean energy transfer rate coefficient}}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v mean energy transferred per collision}}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960
\f20\fs20 {\v energy transferred per collision}}}\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab <{\f23 D}{\i E}> = \|O(\|S\|DO12(\|D\|BA3(){\fs18 0}),\|S\|UP16(\|D\|FO2(){\fs18 \'b0}),{\fs36\dn4 \'ba})({\i E}\|D\|FO1()\endash \|D\|FO1(){\i E'}\|D\|FO1()) {\i P
}({\i E,E'}\|D\|FO1()) d{\i E}\tab (8)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 or the mean-square energy transferred per collision, <{\f23 D}{\i E}{\fs18\up6 2}>:{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v mean-square energy transferred per collision}}{\xe
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v root mean-square energy transfer rate coefficient}}\par
}\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab <{\f23 D}{\i E}{\fs18\up6 2}>= \|O(\|S\|DO12(\|D\|BA3(){\fs18 0}),\|S\|UP16(\|D\|FO2(){\fs18 \'b0}),{\fs36\dn4 \'ba})({\i E}\|D\|FO1()\endash \|D\|FO1(){\i E'}\|D\|
FO1()){\fs18\up6 2} {\i P}({\i E,E'}\|D\|FO1()) d{\i E}\tab (9)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 or the average {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 downward energy transferred per collision}}:\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab <\'c6{\i E}{\fs14\dn4 down}> = \|F(\|I({\fs18\dn4 0}{\i ,E'}{\fs14 , }({\i E'} \endash {\i E}) {\i P}({\i E,E'}\|D\|FO1()) d{\i E),\|I(}{\fs18\dn4 0}{\i ,E'}{
\fs14 , }{\i P}({\i E,E'}\|D\|FO1()){\i dE)) }\tab (10)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 (see Note A). The falloff curve is sensitive largely (but not completely) only to the value of any one of these quantities (along with {\i\f23 w}), rather than the full functional form of {\i P}({\i
E,E'}\|D\|FO1()). Hence to a moderately good approximation, it is only necessary to specify, for example, {\i\f23 w} and <\'c6{\i E}{\fs18\dn4 down}> to calculate a falloff curve, given {\i k}({\i E}
). However, there is some sensitivity to the functional form, and one should endeavor to use as valid a form for {\i P}({\i E,E'}\|D\|FO1()) as possible. Suitable such forms include the exponential-down model:\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab {\i P}({\i E,E'}\|D\|FO1()) = \|F(1,{\i N}({\i E'}\|D\|FO1())) exp\|B\|BC\|[(\endash \|B(\|F({\i E'}\|D\|FO1()\endash \|D\|FO1(){\i E},{\i\f23 a}))) ({\i E}<{\i
E'}\|D\|FO1())\tab (11)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 [where {\i N}({\i E'}) is a normalizing factor so that eq 4 is obeyed], or the form based on the {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 biased random walk model}}
[Lim K.F. and Gilbert R.G. (1990), {\i J. Chem. Phys}, {\b 92}, 1819]:\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 {\i \tab \tab P}{\scaps\fs14\dn4 BRW}({\i E,E'}\|D\|FO1()) = (4\'b9{\i s}{\fs18\up6 2}){\fs18\up6 \endash }{\fs16 \|S\|UP4(1)\|S\|UP3(/)\|S\|UP2(2)} exp\|B(\|F(\endash ({\i z
}s{\fs18\up6 2} + {\i E} \endash {\i E'}){\fs18\up6 2},4{\i s}{\fs18\up6 2}))\tab (12)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 where the quantity {\i s} has the dimensions of energy and:\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 {\i \tab \tab z} = \endash \|F(\'b6ln{\i f}({\i E}),\'b6{\i E})\tab (13)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 where {\i f}({\i E}) = {\i\f23 r}({\i E}) exp(\endash {\i E}/{\i k}{\fs16\dn2 B}{\i T}). The program also allows user specification of other forms for {\i P}({\i E,E'}\|D\|FO1()) (Note I).\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 The program also takes account of {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 angular momentum conservation}i.J-conservation;} in the falloff and low-pressure regimes. Either one can use the
\ldblquote initial-J-independent\rdblquote model for rotational energy transfer, which is expected to be valid for most cases of interest, or one must input a value for the average downward rotational energy transferred per collision, <{\f23 D}{\i R}{
\fs18\dn4 down}>.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960
A simple means of calculating a value of the average energy transfer for an arbitrary reactant, bath gas and temperature is not yet available, although some theories are beginning to emerge. The biased random walk model is a readily-applied semi-empirical
means of estimating this quantity, which has acceptable reliability for a range of systems.{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v biased random walk model}}}
This can be implemented using the program BRW (also see Note I); the program accepts the value of the parameter {\i s} of eq 12 directly, and outputs the corresponding values of <\'c6{\i E}{\fs18\dn4 down}>, <\'c6{\i E}> and <\'c6{\i E}{\fs18\up6 2}>{
\fs18 \|S\|UP4(1)\|S\|UP3(/)\|S\|UP2(2)}. Other methods of finding average energy transfer values have been discussed by Gilbert and Smith.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b The master equation step}
The solution of the master equation, eq 5, and normalization, eq 4, are carried out using methods discussed in detail in Gilbert and Smith. In brief, eq 5 is solved by replacing it with an appropriate finite difference equation (by replacing the integrati
on over d{\i E} by a sum with a \ldblquote grain size\rdblquote {\f23 d}{\i E}), so that the problem becomes equivalent to an eigenvalue calculation. The eigenanalysis is performed by a special variant of the \ldblquote Nesbet\rdblquote
algorithm. Two alternative means of implementing {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 angular momentum conservation}} in the master equation solution are employed: (i)\~the \ldblquote initial-J-independent\rdblquote
treatment (valid for any tight transition state, and for simple fission transition states involving uncharged species) and (ii)\~the exponential-down model for rotational energy transfer (required for an {\v {\xe\pard\plain
\qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 ion-molecule reaction}} and perhaps some simple-fission transition states involving small uncharged species). The latter requires a value of the average downward rotational energy transfer per collision, <{\f23 D}
{\i R}{\fs18\dn4 down}>. When full angular momentum conservation is used: see note C(ix).{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v angular momentum conservation}}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v
J-conservation}}}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 {\v {\tc\tcl1\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 PROGRAM STRUCTURE}}\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 The FALLOFF package includes two separate main programs. The first, called \ldblquote RRKM{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v RRKM theory}}}\rdblquote
, carries out the RRKM calculation, computes high pressure rate parameters and also gives strong collision values for both {\i k}{\fs18\dn4 uni}
and for the low-pressure rate coefficient. It also generates a file (on unit 10) containing all data which will be subsequently used in the solution of the {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 master equation}}
, eq 5. This data consists of lists of {\i\f23 r}({\i E}) and {\i k}({\i E}) (or of J-averaged {\i k}({\i E,J}) for each temperature and pressure, if J conservation is used{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v
angular momentum conservation};{\v .i.J-conservation}}}) and suitable values for all parameters required in the master equation solution. The second program, called \ldblquote MASTER\rdblquote
, solves eq 5. It also computes the low-pressure limiting rate coefficient, the collisional efficiency ({\i\f23 b}), the \ldblquote Lindemann-Hinshelwood\rdblquote factor ({\i F}{\fs14\dn4 LH}) and the strong ({\i F}{\fs14\dn4 SC}) and weak ({\i F}{
\fs14\dn4 WC}) collisional broadening factors (see Note J). It can also be used to compute a complete falloff curve around a median pressure.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\par
\pard \qc\ri29\sl280\tx720\tqc\tx3960 {\b INPUT} \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 Unless explicitly stated, all input is in free format [i.e., READ (5,*) instead of a format number; the user can change to formatted input if desired].{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v
input description}}}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 {\v {\tc\tcl1\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 PROGRAM RRKM{\v .i.RRKM theory}}} \par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 The output from this program is on two files: on unit 6 one finds the usual printed output, while on unit 10 one has a data file [containing inter alia lists of {\i k}({\i E}) and {\i\f23 r}({\i E}
)] which serves as the input file for MASTER.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
{\b \tab \tab VARIABLES AND THEIR DESCRIPTION}\par
\par
1.\tab {\b TITLE}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 Title; up to 80 characters.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
2.\tab {\b NN,INC,NP,NT,NCHAN,JAV}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 NN: no. of energy increments to be used (Note D);\par
INC: integration increment in eq 7, in cm{\fs18\up6 \endash 1} (Note E); usually 100 cm{\fs18\up6 \endash 1};\par
NP: no. of input pressures;\par
NT: no. of input temperatures;\par
NCHAN: no. of reaction channels (up to 3);\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 JAV: flag; if JAV = 0, includes {\v {\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 angular momentum conservation}{\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160
\f20\fs20 {\v J-conservation}}}
only in high-pressure limit (use for tight transition states); if JAV = 1, full angular momentum conservation, including the falloff regime in file passed on to MASTER (use for loose transition states of neutrals); if JAV = 2, as for JAV=1, but here inter
nal rotations in transition state are hindered by a sinusoidal potential whereas for JAV = 1 they are treated as free rotors (use for simple fission transition states of neutrals or of ions; Note H). JAV > 0 only for {\v {\xe\pard\plain
\s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 simple-fission{\v transition state}}} transition states (neutral radical-radical or {\v {\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 ion-molecule reaction}}).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
{\i Inputs numbers 3-7 present only if JAV=2\par
}\par
3.\tab {\b IHIND(IN), ION(IN), IN = 1, NCHAN}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\plain \f20 | }IHIND {\plain \f20 |} = no. 2-dimensional sinusoidally-hindered rotors in transition state (one or two); if IHIND < 0, hard-sphere steric interactions also included;\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 ION: flag; if ION = 0, neutral radical-radical reaction; if ION = 1, ion-molecule reaction\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
{\i Inputs numbers 4 to 7 are looped for IN = 1,NCHAN, and are only present for IHIND non-zero}\par
4.\tab {\b ROTINC(IN),ERR(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 ROTINC: rotational energy grain size (cm{\fs18\up6 \endash 1}), typically 100 cm{\fs18\up6 \endash 1};\par
ERR: convergence parameter, typically 2.5{\f23 \'b4}10\endash 3\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
{\i Inputs numbers 5 and 6 only if IHIND(IN) < 0, ie there are hard-sphere steric interactions}\par
\par
5. \tab {\b THETA1(IN),ISYM1(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 THETA1: angle (degrees) through which hindered rotor can move; 0\'b2{\i\f23 q}\'b2180.\par
ISYM1: symmetry number of hindered rotor.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
6. {\i (only present if IHIND(IN) = \endash 2)} {\b THETA2(IN),ISYM2(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 as in input number 5, for second hindered rotor.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
7.\tab {\b GAMMA(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 GAMMA(IN) = value of <{\f23 D}{\i R}{\fs18\dn4 down}> (cm{\fs18\up6 \endash 1}); if GAMMA > 600 cm{\fs18\up6 \endash 1}, \ldblquote initial-J-independent (strong collision in J)
\rdblquote model used for J-conservation{\v {\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\v angular momentum conservation}}{\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\v J-conservation}}}
in falloff regime; if GAMMA < 600 cm{\fs18\up6 \endash 1}, exponential-down rotational energy transfer collision solution of J-conserving master equation. GAMMA > 600 cm{\fs18\up6 \endash 1} is the most common case.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\i Inputs 8n to 10n are for the case of JAV > 0 and ION=0: neutral radical-radical }{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\i simple-fission transition state;}{\i\v }{\i
Inputs 11n to 20n are for any JAV except the case of ion-molecule reactions (i.e., 11n-20n for neutral tight and loose transition states). Inputs 8i to 24i only if JAV > 0 and ION = 1\: ion-molecule transition states.}{\i\v .i.ion-molecule reaction}}}{
\i \par
}\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
8n.\tab {\b RCPL(IN),REQ(IN)}, IN = 1,NCHAN\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 RCPL: separation of fragments in transition state (\'81);\par
REQ: equilibrium bond length (\'81). See Note C(ix).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
{\i Inputs nos. 9 and 10 looped through I = 1,NCHAN. See Note C(ix).}\par
\par
9n.\tab {\b NV(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 number of potential points used to fit a Morse curve to {\i V}({\i r}\|D\|FO1()), the interaction between the A and B moieties in {\v {\xe\pard\plain
\s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 simple-fission transition state}}.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
10n.\tab {\b RVCH(IN,II),VCH(IN,II)}, II = 1,NV(IN)\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 Values of {\i r } (\'81) and {\i V}({\i r}\|D\|FO1()) (kcal mol{\fs18\up6 \endash 1})\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
11n.\tab {\b (JF(IN), IN = 1,NCHAN); NF}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 JF(IN): number of distinct frequencies in transition state for channel IN;\par
NF: number of distinct frequencies in reactant molecule.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
12n.\tab {\b E0K(IN)}, IN = 1,NCHAN\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 critical energy {\i E}{\fs14\dn4 0} (kcal mol{\fs18\up6 \endash 1}) for each channel. Must be in increasing order.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
13n.\tab {\b (SRC(IN), IN = 1,NCHAN), SRM}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 SRC(IN): {\i\f23 s}/{\i n} for IN{\fs18\up6 th} activated complex ({\i\f23 s} =symmetry number; {\i n} = no. of optical isomers);{\v {\xe\pard\plain
\s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\v symmetry number}}{\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\v reaction path degeneracy}}}\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 SRM: same, for reactant. See Note G.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
14n.\tab {\b (BCMPLX(IN), IN = 1,NCHAN), BMOLEC}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 BCMPLX(IN): rotational constant (cm{\fs18\up6 \endash 1}) for inactive external rotations for IN{\fs18\up6 th} activated complex;\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 BMOLEC: same for reactant. See Note C(i)-(viii).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
15n.\tab {\b (N(IN), IN = 1,NCHAN), NINTR}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 N(IN): no. of active rotations (internal plus external) in IN{\fs18\up6 th} activated complex; if input as negative number, implies linear activated complex.\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 NINTR: same for reactant. See Note C.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
{\i Input no. 16n looped through IN = 1,NCHAN\par
}\par
\pard \qj\fi-720\li720\ri29\sl280\tx720\tqc\tx3960 16n.\tab {\i (only present if N(IN) > 0) } {\b (BVEC(IN,J),SIGVC(IN,J),IRTDMC(IN,J), J\~=\~1,N(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 for J{\fs18\up6 th} active rotation of IN{\fs18\up6 th} activated complex:\par
BVEC(IN,J): rotational constant (cm{\fs18\up6 \endash 1});\par
SIGVC(IN,J): {\i\f23 s}/{\i n};\par
IRTDMC(IN,J): dimension.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\fi-720\li720\ri29\sl280\tx720\tqc\tx3960 17n.\tab {\i (only present if NINTR > 0) }{\b BVECM(J),SIGVCM(J),IRTDMM(J)}, J = 1,NINTR\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 as in input 16n, except for J{\fs18\up6 th} active rotation of reactant\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
18n.\tab {\b SGMA,WT1,WT2,EPS}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 SGMA: hard-sphere or Lennard-Jones diameter (\'81) for reactant-bath-gas interaction, for calculating {\i\f23 w};{\v {\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160
\f20\fs20 {\v collision frequency}}}\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 WT1: molecular weight of reactant (a.m.u.). If negative and JAV{\f23 \'b9}
0 then used as a flag for entering a range of inactive two-dimensional external rotational constants vs breaking bond length, this option should be used if the centre of mass of at least one of the fragments does not coincide with the
pivot atom of that fragment (see note C);\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 WT2: molecular weight of bath gas (a.m.u.);\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 EPS: if negative, flag for using hard-sphere collision frequency; if positive, Lennard-Jones well depth {\i\f23 e} (K), for calculating {\i\f23 w}.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
{\i Inputs nos.19n and 20n are looped through IN=1,NCHAN and are only present if WT1}{\i\f23 <}{\i 0 and JAV}{\i\f23 \'b9}{\i 0}\par
\par
19n.\tab {\b NB(IN)}\par
\tab number of input pairs of breaking bond length and rotational constant for reaction\par
\tab channel IN\par
\par
20n. \tab {\b RR(IN,I),BB(IN,I)},I=1,NB(IN)\par
\tab RR(IN,I): I{\up6 th} entered bond length of breaking bond along IN{\up6 th} reaction channel\par
\tab BB(IN,I): corresponding rotational constant (cm{\up6 -1}) for inactive two-dimensional\par
\tab external rotation.\par
\par
{\i Input no. 21n looped through IN = 1,NCHAN\par
}\par
21n.\tab {\b NC(IN,I),JC(IN,I)}, I = 1,JF(IN)\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 NC(IN,I): vibrational frequency (cm{\fs18\up6 \endash 1}) of I{\fs18\up6 th} oscillator of IN{\fs18\up6 th} activated complex;\par
JC(IN,I): degeneracy of I{\fs18\up6 th} oscillator of IN{\fs18\up6 th} activated complex.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
22n.\tab {\b NM(I),JM(I)}, I = 1,{\caps nchan}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 as in input no. 21n, for reactant.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\i Inputs 8i - 24i are only for }{\i\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\i ion-molecule reaction}}}{\i s}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
8i.\tab {\b BMOLEC}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 rotational constant (cm{\fs18\up6 \endash 1}) for inactive two-dimensional external rotation of reactant (\ldblquote collision complex\rdblquote ) (see Note C).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
9i.\tab {\b NINTR}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 no. of active (internal plus external) rotors of reactant (\ldblquote collision complex\rdblquote ); negative implies linear molecule.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
10i.\tab {\b BVECM(I),SIGVCM(I),IRTDMM(I)}, I = 1,NINTR\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 for I{\fs18\up6 th} active rotation of reactant (\ldblquote collision complex\rdblquote ):\par
BVECM(I): rotational constant (cm{\fs18\up6 \endash 1});\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 SIGVCM(I): {\i\f23 s}/{\i n} ({\i\f23 s} = symmetry number; {\i n} = no. of optical isomers);{\v {\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\v symmetry number}}{\xe
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\v reaction path degeneracy}}}\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 IRTDMM(I): dimension.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
11i.\tab {\b NF}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 no. of distinct frequencies of reactant (\ldblquote collision complex\rdblquote ).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
12i.\tab {\b NM(I),JM(I)}, I = 1,NF\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 frequency (cm{\fs18\up6 \endash 1}) and degeneracy for I{\fs18\up6 th} vibration of reactant (\ldblquote collision complex\rdblquote ).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
13i.\tab {\b KCAPT,WT1}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 KCAPT: capture rate coefficient (cm{\fs18\up6 3} s{\fs18\up6 \endash 1}) of reactant (\ldblquote collision complex\rdblquote
) and bath gas; see eqs 5.5.19 and 5.5.20 of Gilbert and Smith;\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 WT1: molecular weight of reactant (\ldblquote collision complex\rdblquote ) (a.m.u.).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
{\i Inputs 14i - 22i are looped through for IN = 1,NCHAN\par
}\par
14i.\tab {\b DH0(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\f23 D}{\i H}{\fs14\dn4 0}{\f23 \'b0} (dissociation energy) for IN{\fs18\up6 th} channel (kcal mol{\fs18\up6 \endash 1}).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
15i.\tab {\b REQ(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 equilibrium bond length (\'81) for IN{\fs18\up6 th} channel.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
16i.\tab {\b RCPL(IN),BCMPLX(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 RCPL(IN): separation of fragments (\'81) in IN{\fs18\up6 th} transition state;\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 BCMPLX(IN): rotational constant (cm{\fs18\up6 \endash 1}) for external inactive two-dimensional rotation of IN{\fs18\up6 th} transition state.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
17i.\tab {\b DIP(IN),POLZ(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 for neutral moiety (i.e., B in ion-molecule reaction A{\fs18\up6 +} + B {\f23 \'ae} AB{\fs18\up6 +}){\v {\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\v
ion-molecule reaction}}\par
}\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 DIP(IN): dipole moment (Debye);\par
POLZ(IN): polarizability (cm{\fs18\up6 3}).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
18i.\tab {\b WTA(IN),WTB(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 molecular weights (a.m.u.) for A and B moieties as in input no. 17i.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
19i.\tab {\b N(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 no. active (internal plus external) rotors in activated complex (if negative, linear transition state)\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
20i.\tab {\b BVEC(IN,II),SIGVC(IN,II),IRTDMC(IN,II)}, II = 1,N(IN)\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 as for input no. 10i except for II{\fs18\up6 th} active rotor of IN{\fs18\up6 th} activated complex\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
21.\tab {\b JF(IN)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 no. of distinct frequencies of transition state\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
22.\tab {\b NC(IN,II),JC(IN,II)}, II = 1,JF(IN)\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 as for input no. 12i except for II{\fs18\up6 th} frequency and degeneracy of IN{\fs18\up6 th} activated complex.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
23i.\tab {\b JRECOM}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 flag: if JRECOM = 0, dissociation; if JRECOM = 1, association.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
{\i Inputs nos. 24i and 25i only if JRECOM = 1\par
}\par
24i.\tab {\b INCHAN}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 flag: indicates channel (1, \'c9, NCHAN) through which association proceeds.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
25i.\tab {\b RADST}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 radiative stabilization constant (cm{\fs18\up6 \endash 1}); if such stabilization is insignificant, a value of 10{\fs18\up6 \endash 10} cm{\fs18\up6 \endash 1} indicates this.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
{\i Remaining inputs for all cases}\par
\par
26.\tab {\b PRESS(I)}, I = 1,NP\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 list of pressures (torr).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
27.\tab {\b TEMP(I)}, I = 1,NT\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 list of temperatures (K); must be in decreasing order.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 \page {\v {\tc\tcl1\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 PROGRAM MASTER }}\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20
The basic input file for this program (with suitable values for all parameters) is prepared (on unit 10) by program RRKM. In using MASTER, one would typically only need to change the following: NALPHA, ALPHAV, NP, PR (however, if {\v {\xe\pard\plain
\qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 angular momentum conservation}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v J-conservation}}} is employed, these last two must not be changed from those used in the program RRKM{\v .}
to prepare the master equation data file). This file prepared by RRKM contains {\i inter alia} the {\i\f23 r}({\i E}) and the {\i k}({\i E}) [or, if the falloff J-conserving option is used, the J-averaged {\i k}({\i E}
) for each pressure and temperature]. Most of the input data from RRKM must not be changed in MASTER; the only parameters which can be changed are indicated below with an asterisk (*) (may be changed under any circumstances) or a hash (#) (may be changed o
nly if the falloff angular momentum conservation{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v angular momentum conservation}}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v J-conservation}} }
option is not used). Parameter values which may be changed are in free format (except for TITLE).{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v input description}}}
Note: there are two versions of MASTER supplied: MAS55 and MAS77. MAS55 is used for all except ion/molecule reactions, for which one uses MAS77.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\tab {\b \tab VARIABLES AND THEIR DESCRIPTION}\par
\par
1.\tab *{\b TITLE}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 Title; up to 80 characters.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
2.\tab {\b INC,NCHAN,INCCHK}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 INC,NCHAN: as in RRKM (Note B);\par
INCCHK = value of INC used in RRKM (aborts if INCCHK \'ad 100; Note D).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
3.\tab *{\b ERR1,ERR2,ERR3}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 ERR1 = tolerance for truncation of maximum energy considered (i.e., the value of \ldblquote infinity\rdblquote in the upper bound of the integrations);\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 ERR2 = tolerance for eigenvalue convergence (Note K);\par
ERR3 = tolerance for truncation of {\i P}({\i E,E'}\|D\|FO1()) (Note A).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
4.\tab {\b E0}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 critical energy {\i E}{\fs14\dn4 0} (or lowest critical energy if a {\v {\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 multichannel reaction}}), in kcal\~mol{
\fs18\up6 \endash 1}. If J-conservation{\v {\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\v angular momentum conservation}}{\xe\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 {\v J-conservation}}}
option used, E0 is lowest rotational barrier.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
5.\tab *{\b NALPHA}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 no. of values of {\i\f23 a} (see Note I).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
6.\tab *{\b ALPHAV(I)}, I=1,NALPHA\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 array of {\i\f23 a} values (cm{\fs18\up6 \endash 1}); see Note I.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
7.\tab *{\b IXV,JXV}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 Parameters for functional form of {\i P}({\i E,E'}\|D\|FO1()) (see Note I).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
8.\tab #{\b NP}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 number of pressures.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
9.\tab #{\b PR(I)}, I=1,NP\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 array of pressures, in torr (Note J).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 10.\tab *PALMT\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 parameter for eigenvector calculation: all {\i g}({\i E}) for {\i E} < {\i E}{\fs14\dn4 0} {\f23 \'b4} PALMT assigned their equilibrium value (Note K).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
11.\tab {\b JAV}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 See RRKM input.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
12.\tab {\b NTEMP}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 no. of input temperatures.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
13.\tab {\b TEMPV(I)}, I = 1,NTEMP\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 array of input temperatures (K).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
14.\tab #{\b SGMA,WT1,WT2,EPS}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 See RRKM input.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
15.\tab *{\b IOPTHT,IOPTPR}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 IOPTHT: if IOPTHT = 0, does low-pressure limit as well as calculation at input pressures (required to obtain collision efficiency b). If IOPTHT \'ad
0, only does calculation at input pressures.\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160
IOPTPR: parameter for criterion for eigenvalue convergence. If IOPTPR = 1, convergence uses only total rate for first two channels (used always for NCHAN=1 and can be used for NCHAN > 1 if desired); if IOPTPR = 2, requires convergence in channel 1 and cha
nnel 2 separately (usual value for NCHAN > 1, but IOPTPR=1 can be used for NCHAN>1 if rate coefficients from the various channels are very different in magnitude).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
16.\tab {\b NDEGS}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 no. of {\i\f23 r}({\i E}) values to be read in.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
17.\tab {\b RHO(I)}, I=1,NDEGS\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 array of {\i\f23 r}({\i E}) values.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
18.\tab {\b TEMPV(I),CORRAT(I)}, I=1,NTEMP{\i (only read if JAV = 0)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 TEMPV(I): temperature (repeated from input no. 13);\par
CORRAT(I): correction factor (Note F).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
19.\tab {\b NTRH}{\i (only read if JAV \'ad 0)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 no. of rotational threshold energies.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
20.\tab {\b RATHP,CORAV,CORPF }{\i (only read if JAV \'ad 0)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 RATHP: high-pressure rate coefficient;\par
CORAV,CORPF: average correction factors (Note F).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
21.\tab {\b STLPJ}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 strong-collision low-pressure rate coefficient.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
22.\tab {\b NRATES}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 no. of microscopic rate coefficients {\i k}({\i E}) to be input.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
23.\tab {\b R1(I)}, I=1,NRATES\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 list of {\i k}({\i E}) for channel 1 (or J-averaged {\i k}({\i E}) if JAV\~>\~0).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
24.\tab {\b R2(I)}, I=1,NRATES\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 if NCHAN=1: list of zeroes;\par
if NCHAN > 1: list of {\i k}({\i E}) for other channels (or J-averaged {\i k}({\i E}) if JAV\~>\~0).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
*: parameter value may be changed under any circumstances.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 #: parameter value may be changed only if the falloff {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 angular momentum conservation}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v
J-conservation}}} option was not used to prepare file (i.e., these cannot be changed if file was prepared with JAV \'ad 0).\par
\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 \page {\v {\tc\tcl1\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 PROGRAM GEOM (co-authors\: I G Pitt, P G Greenhill)}}\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20
This calculates all the quantities for rotations required as described in Note C. It takes as input the atomic weights, and either bond angles, bond lengths, dihedral angles and the connectivity (which atom is joined to which) or the Cartesian coordinates
(e.g., obtained from bond angles, etc., using Gaussian 86 program) and outputs Cartesian coordinates and moments of inertia of the whole molecule. The Cartesian coordinates are obtained from the bond and dihedral angles using the method of H.B. Thompson,
{\i J. Chem. Phys.} {\b 47}
, 3407 (1967), wherein a definition of the dihedral atoms can be found; this is explained in detail in Appendix II (see also Note M). The bond angle is defined as that between the current atom, the atom to which it is bonded, and the first atom entered as
being bonded to the latter. The dihedral angle is defined as that between the current bond and the first bond entered for the preceding group. If required, the program also computes the quantities for a Gorin calculation on a {\v {\xe\pard\plain
\qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 simple-fission transition state}}. These latter have been described in detail in Gilbert and Smith. Specifically, if one has a transition state consisting of two moieties, A and B, one states the indexes of the
\ldblquote pivot atoms\rdblquote in each moiety (i.e, the indices of the atoms whose bond breaks). The program then finds the average hard-sphere hindrance angles when each of these moieties is rotated about its pivot, and then
the five rotational constants (incorporating hindrances) that are required for the RRKM{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v RRKM theory}}}
Gorin calculation. These include the moments of inertia of each moiety about its centre of mass. This is particularly convenient for doing a canonical variational transition state calculation{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {
\v variational selection of transition state}}} (see Note N and also Appendix I). The output includes the coordinates in the axes defined by the principal moments of inertia, which can be useful e.g., for the QCPE trajectory program VENUS of Hase.{\v {\xe
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v input description}}}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960
The hard-sphere hindrances calculated by GEOM are found using pre-assigned values of the van der Waals radii of many common atoms: H, D, C, Si, Ge, O, S, Se, N, P, As, F, Cl, Br and I. The atoms are identified by the program based on the input atomic weigh
t. If the user wishes to change these radii, or to use atoms different from those supplied, it is necessary to make appropriate changes in the Fortran code.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\tab \tab {\b VARIABLES AND THEIR DESCRIPTION}\par
\par
1.\tab {\b TITLE}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 Title; up to 80 characters.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
2.\tab {\b NN}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 |NN| = total number of atoms; if NN > 0, input is in terms of bond lengths, angles and dihedral angles; if NN < 0, input as Cartesian coordinates.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\fi-720\li720\ri29\sl280\tx720\tqc\tx3960 3.\tab {\b NA(I),NB(I),A(I),B(I),R(I),MASS(I)},I=1,NN {\i (read in only for the case where NN\~> 0)}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 NA(I) = index of atom I (Note L);\par
NB(I) = index of previous atom to which atom I is bonded;\par
A(I) = bond angle (degrees) between this bond and the last bond of NB(I);\par
B(I) = dihedral angle (degrees); see Appendix II;\par
R(I) = bond length (\'81);\par
MASS(I) = atomic weight of atom I; see Note L.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
4.\tab {\b X(I),Y(I),Z(I),MASS(I)}, I=1,NN( this is read in only for the case where NN < 0)\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 Cartesian coordinates and masses of each atom.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
5.\tab {\b ITEST}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 If ITEST = 0, finds Cartesian coordinates and moments of inertia only; if ITEST\~\'ad
0, also finds hard-sphere hindrance angles and rotational constants for simple-fission transition state.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
{\i Remaining data only read if ITEST \'ad 0}\par
\par
6.\tab {\b NATOMA,INDEXA}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 NATOMA = no. of atoms in moiety A; if NATOMA < 0, code accepts input of van der Waals radii for specified atoms, rather than default values \par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 INDEXA = index of pivot atom in moiety A.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
7.\tab {\b LISTA(I)}, I=1,NATOMA\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 Indexes of atoms in moiety A.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
8.\tab {\b INDEXB}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 Index of pivot atom in moiety B.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
9.\tab {\b IVAN\par
}\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 number of van der Waals radii to be input\par
\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 \par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 10.\tab {\b LATR(I), ATVR(I)}, I = 1,IVAN\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 LATR(I) = index of atom for which radius input is required\par
ATVR(I) = radius\par
\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 \page {\v {\tc\tcl1\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 PROGRAM BRW}}\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 This calculates the average energy transfer, required for falloff and low-pressure calculations, using the \ldblquote {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 biased random walk{
\v model}}}\rdblquote model B [K F Lim and R G Gilbert, {\i J. Chem. Phys}, {\b 92}, 1819 (1990)]. This approximate treatment enables <{\f23 D}{\i E}
> or any other measure of the average energy transferred per collision to be estimated from a knowledge of the physical properties of the reactant and bath gas. It should be of acceptable reliability for monatomic and diatomic bath gases. However, it is em
phasized that such {\i a priori} calculations are still in their infancy, and this model should only be used if no experimental data are available for the requisite system. See Gilbert and Smith for an extensive discussion.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960
The basic assumption of the model is that energy exchange between the bath gas and the highly excited reactant is pseudo-random during the collision: an assumption that is verified by trajectory calculations. Imposing the condition of microscopic reversibi
lity then leads to the functional form of eq 12 for {\i P}({\i E,E'}\|D\|FO1()). The most important characteristic of the dynamics is that the magnitude of the energy change is governed by many atom
-atom interactions. Making a series of assumptions about the details of the dynamics (each assumption being tested against trajectory data) leads to the formulae given in eqs 32 to 40 and 43 of Lim and Gilbert. This yields a readily evaluated expression fo
r the value of the parameter {\i s} in eq 12; this value can in turn be directly input into MASTER as described in Note I, and thence a falloff curve, and average energy transfer quantities, calculated.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 The input required consists of (i) masses and Lennard-Jones parameters of reactant and bath gas, and (ii) local \ldblquote atom-atom\rdblquote
Lennard-Jones potential parameters, the latter determined as described in Lim and Gilbert and in Gilbert and Smith. The local Lennard-Jones parameters can either be chosen by the user or \ldblquote optimal\rdblquote values used as a default.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 Because the model is an approximate one, it can give unphysical values. For this reason, it gives a warning message if {\i s} is too large.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\tab \tab {\b VARIABLES AND THEIR DESCRIPTION}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v input description}}}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\i Input nos. 2 to 5can be repeated as often as desired for successive calculations; the program terminates when given a negative value of EPS in input no. 2.\par
}\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\i \par
}1.\tab {\b TITLE}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 Title; up to 80 characters.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
2.\tab {\b EPS,SIGMA,M,T,NATOMS,MLIGHT\par
}\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 EPS: Lennard-Jones {\i\f23 e} for reactant/bath gas (K);\par
SIGMA: Lennard-Jones {\i\f23 s} for reactant/bath gas (\'81);\par
M: reduced mass of reactant and bath gas (a.m.u.);\par
T: temperature (K);\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 NATOMS: no. of distinct types of atoms in reactant (e.g., for C{\fs18\dn4 3}H{\fs18\dn4 7}Br, there are 3 types of atoms: C, H and Br);\par
MLIGHT: atomic weight of lightest atom in reactant (a.m.u.) (usually MLIGHT = 1.008, the atomic weight of hydrogen).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
3.\tab {\b EPSBG,SIGBG,MBG}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 EPSBG: Lennard-Jones {\i\f23 e} for bath gas (K);\par
SIGBG: Lennard-Jones {\i\f23 s} for bath gas (\'81);\par
MBG: atomic weight of bath gas (a.m.u.).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\i Input no. 4 is looped over 1 to NATOMS; each refers to a distinct type of reactant atom (e.g., one line for the values for C in C}{\i\fs18\dn4 3}{\i H}{\i\fs18\dn4 7}{\i Br, then another for H and another for Br)
\par
}\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
4.\tab {\b MM,NST,EPSL,SIGL}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 MM: mass of reactant atom;\par
NST: its stoichiometric coefficient (e.g., if this is for C in C{\fs18\dn4 3}H{\fs18\dn4 7}Br, NST=3);\par
\pard \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 EPSL: local Lennard-Jones {\i\f23 e} for this atom (units: K; if negative, default value for this atom is used; default values available for H, D, C, N, O, F, Cl, S, Br and I);\par
SIGL: local Lennard-Jones {\i\f23 s} for this atom (units: \'81; if negative, default value for this atom is used; default values available for H, D, C, N, O, F, Cl, S, Br and I).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
5.\tab {\b ECOLL,NVIB,NU}\par
\pard\plain \s11\qj\fi-720\li2160\ri29\sl280\tx720\tx2160 \f20\fs20 ECOLL: internal energy of reactant (cm{\fs18\up6 \endash 1}) (not used, present for consistency with earlier version);\par
NVIB: no. of vibrational frequencies of reactant (not used, present for consistency with earlier version);\par
NU: highest frequency (cm{\fs18\up6 \endash 1}) of reactant (usually {\i ca.} 3000 cm{\fs18\up6 \endash 1}, that of a C\_H stretch).\par
\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 \page APPENDIX I\par
The high-pressure limit for {\v {\xe\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 simple-fission transition states}}\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v variational selection of transition state}}\par
}\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 {\v {\tc\tcl1\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 {\v APPENDIX I (high-pressure limit - simple fission)}}\par
}\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 The high pressure limit of the rate coefficient {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}) is determined as the Boltzmann average of {\i k}({\i E,J}) [{\i E}
is the internal energy of the molecule over the equilibrium population distribution]. It is calculated using eq 3.8.29 of Gilbert and Smith. This counts all possibly reactive flux passing through the position {\i r}{\fs18\up6 \'a0}
on the reaction coordinate, discounting only that which corresponds to molecules without enough total energy to surmount the centrifugal barrier for a given J. IT IS THEREFORE SENSITIVE TO THE POSITION OF THE TRANSITION STATE, {\i r}{\fs18\up6 \'a0}
, AND MAY BE USED VARIATIONALLY TO DETERMINE THE BEST VALUE FOR {\i r}{\fs18\up6 \'a0}: see Note N. This will be the position producing the minimum {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}
) (canonical variation). As a means of semi-automating the canonical variational process, program RRKM{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v RRKM theory}}} also calculates {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14
\'b0}) by the alternative method involving the {\i f}{\fs18\dn4 \'b0} factor of Forst (e.g., eqs 3.8.33 and 3.8.34 of Gilbert and Smith), which assumes that a) the transition state always lies at the centrifugal barrier, and b) {\i\f23 r}{\fs18\up6 \'a0
}(E) does not vary with the position {\i r}{\fs18\up6 \'a0} of the transition state. This latter method produces an approximately correct {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}) (which is much less sensitive to the location {\i r}{
\fs18\up6 \'a0} of the transition state). The high pressure limits calculated by the two different methods will essentially agree (say, within 20%) only if the transition state is placed at a position {\i r}{\fs18\up6 \'a0}
which is an acceptable average over the contributing angular momenta. If {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}
) calculated by the two different methods deviates by more than 20%, the program will output a warning message suggesting that the transition state be moved to a different {\i r}{\fs18\up6 \'a0} which provides better agreement. \-
Note that for the J-conserving option an appropriate numerical correction factor is determined by evaluating {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}) via a complete numerical integration of {\i k}({\i E,J}
) and comparing the result with that of eq 3.8.29 of Gilbert and Smith.{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v angular momentum conservation}}}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 {\v {\tc\tcl1\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 {\v A}{\caps\v PPENDIX II}{\v (dihedral angles)}}}\par
\pard \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 A{\caps ppendix} II\par
Dihedral angles\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 Working out {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 dihedral angles}} is complex. To assist users, we present a de
tailed derivation of these for three situations: chloroethane, the chloroethane elimination four-centre transition state, and the methyl/methyl {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v recombination (association) rate coefficient}
}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v simple-fission transition state}} }
simple-fission transition state. In addition, a wide variety of molecules have been included in the sample data sets; data sets for most molecules can then be constructed from these.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
{\b Chloroethane\par
}\tab \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 For simplicity, we will assume that all bond angles are 109\'a1, and standard bond lengths: (C\_C)=1.54 \'81, (H\endash C)=1.1 \'81, (C\endash Cl)=1.77 \'81
. We shall describe how one may specify the structure of the molecule using bond angles, dihedral angles and bond lengths as required by GEOM. Number the atoms in the molecule in the order in which we intend to e
nter them. For instance, chloroethane might be numbered as in Figure 2.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \tab {{\pict\macpict\picw306\pich116
01cb0009001f007d0151110101000a0000000002d0021c08000a220041006ee3002200440045f91422003a00770aeb71001e004400770057007c00440077004400780057007c0057007900440077220045007a0100220047007e01ff220048008201ff22004a008601fe22004b008a01fe71001e00330033003b0045003a00
45003b00450036003300330033003a004522003600430100220033004201ff22002f0041020022002c004002ff220028003f030022004c00e1f70f22004400ef200071001e003500d4004000e1003f00e1004000e1003800d4003500d6003f00e122003b00e0010022003800e0010022003500e0010022003200e001002200
2f00e0020022002b0124f81171001e004a0120005f012b004a0120004a0121005d012b005f0128004a012022004b012400ff22004c012801ff22004d012d00fe22004e013101fe22004f013600fd0300140d000a2800260124032831292800430072014328004300440143280023008102436c28006200390148293f014828
0027003b01482b542c0148280033002a01482bb8160328352928006600ce0328362928004701150328322928003300ca0328372928002b00d9032838292b513e03283329280053013803283429ff}}\par
\tab \par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Fig. 2. } A sample numbering scheme for specifying bond angles, dihedral angles and bond lengths for chloroethane.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 There will be several ways of specifying the structure. It will
be simplest when using the present technique to begin with a terminal atom as in Figure 2. For each atom, we specify another atom (previously entered) to which it is bonded, a bond length, a bond angle, and a dihedral angle. The set of input data for chlo
roethane is given in Table 1.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Table 1.} Bond angles, dihedral angles and bond lengths to specify the structure of chloroethane.\par
\pard\plain \s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \par
\pard\plain \s10\qj\li720\ri331\sl280\brdrt\brdrs \tx720\tqc\tx1260\tqc\tx2520\tqc\tx3870\tqc\tx5040\tqc\tx5940\tqc\tx7020 \f20\fs18 \par
\tab Index for\tab Index for\tab Bond\tab Dihedral\tab Bond \tab Mass of\par
\tab present atom\tab attached atom\tab angle\tab angle\tab length\tab present atom\par
\pard\plain \s12\qj\li720\ri331\sl280\brdrb\brdrs \brdrbtw\brdrs \tx720\tqr\tx1260\tqr\tx2520\tqr\tx3960\tqr\tx5220\tqr\tx6030\tqr\tx7200 \f20\fs18 \par
\tab 1\tab 0\tab 0\tab 0\tab 0\tab 35.45\par
\tab 2\tab 1\tab 0\tab 0\tab 1.77\tab 12.0\par
\tab 3\tab 2\tab 109\tab \endash 120\tab 1.1\tab 1.0\par
\tab 4\tab 2\tab 109\tab 120\tab 1.1\tab 1.0\par
\tab 5\tab 2\tab 109\tab 0\tab 1.54\tab 12.0\par
\tab 6\tab 5\tab 109\tab 180\tab 1.1\tab 1.0\par
\tab 7\tab 5\tab 109\tab \endash 60\tab 1.1\tab 1.0\par
\tab 8\tab 5\tab 109\tab 60\tab 1.1\tab 1.0\par
\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960
When defining a bond length, one identifies the atom to which the present atom is attached by a second index (the second column of Table 1). A bond angle is defined relative to the present atom and two previous atoms. A dihedral angle is the angle between
two planes, one containing the present atom and two previous atoms, and the other containing three previous
atoms (the planes having two of the atoms in common). The method we describe here has an implicit scheme for counting the previous atoms to be used as references for the bond and dihedral angles. This scheme is indicated for the chloroethane example by th
e branching diagram in Figure 3.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \tab \tab {{\pict\macpict\picw143\pich149
01eeffffffff0094008e1101a00082a10096000c010000000200000000000000a1009a0008fffd0000000d000001000a0000002d000c004a0300140d000a2b2e090731202822302229a0009701000affffffff0094008e22001200360012a10096000c010000000200000000000000a1009a0008fffc00000002000001000a
002d00360039003c2b092d0132a00097a10096000c010000000200000000000000a1009a0008fffd00000002000001000a005a00000066000628006300010133a00097a10096000c010000000200000000000000a1009a0008fffd00000002000001000a005a00360066003c29360134a00097a10096000c01000000020000
0000000000a1009a0008fffd00000001000001000a005a006300660069292d0135a00097a10096000c010000000200000000000000a1009a0008fffc00000002000001000a008700360093003c28009000370136a00097a10096000c010000000200000000000000a1009a0008fffc00000001000001000a00870063009300
69292d0137a00097a10096000c010000000200000000000000a1009a0008fffc00000001000001000a008700870093008d29240138a0009701000affffffff0094008e22003f0036001222003f002ddc1222006c0063001222006c005ae51222006c006c1b1222003f003f2412a00083ff}}\par
\par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Fig. 3.} Branching diagram illustrating the counting scheme used for definition of bond and dihedral angles in the chloroethane molecule. Atom \ldblquote 0\rdblquote
, a dummy atom , is not entered in its own right, but only with reference to the first atom (atom 1).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \tab \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960
A bond angle is given with reference to the present atom and the last two atoms above it in the branching diagram, e.g., when entering a bond angle for atom 5 (carbon), it is implicitly assumed that this refers to the angle between the bonds connecting ato
ms 5, 2, and 1 (i.e., C\endash C\endash Cl). A dihedral angle is the angle between two planes, the first plane containing the last three atoms above in the branching diagram and the second plane containing the present atom and the last tw
o atoms above. For example, the dihedral angle for atom 6 is the angle between two planes: the first containing atoms 5,2,and 1 and the second containing atoms 6, 5, and 2.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 We now explain the entries for Table 1 in some more detail. Looking along the Cl\endash C bond, one sees the following projection (Figure 4):\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \tab \tab {{\pict\macpict\picw180\pich202
0934007d006c014701201101a00082a0008e01000a0000000002d00240980018007d006800b90120007d006c00b90120007d006c00b90120000002e90002e90002e90002e90002e90002e90002e90002e90006f60000fff50008f700020700e0f60008f70002180018f60008f7000230000cf60008f70002400002f60008f7
0002800001f6000af8000001fe000080f7000af8000002fe000040f7000af8000006fe000060f7000af8000004fe000020f7000af800000cfe000030f7000af8000008fe000010f7000af8000008fe000010f7000af8000010fe000008f7000af8000410003c0008f7000af800041000200008f7000af800041000380008f7
000af800041000040008f7000af800041000040008f7000af800041000240008f7000af800041000180008f7000af8000010fe000008f7000af8000008fe000010f7000af8000008fe000010f7000af800000cfe000030f7000af8000004fe000020f7000af8000006fe000060f7000af8000002fe000040f7000af8000001
fe000080f70008f70002800001f60008f70002400002f60008f7000230000cf60008f70002180018f60008f700020700e0f60006f60000fff50002e90002e90006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f500
06f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50098001800b9006800f5012000b9006c00f5012000b9006c00f50120000006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50008f7000203ffe0f60008f700023c
081ef6000af8000401c00801c0f7000af800040600080030f7000af80004180008000cf7000af800046000080003f7000bf80005800008000080f8000cf9000603000008000060f8000cf9000604000008000010f8000cf9000608000008000008f8000cf9000610000008000004f8000cf9000620000008000002f8000cf9
000640000008000001f8000df9000380000008fe000080f9000efa000001fe000008fe000040f9000efa000003fe000008fe000060f9000efa000002fe0000fffe000020f9000efa00080400000f00f0000010f9000efa000808000038001c000008f9000efa0008080000600006000008f9000efa00081000018000018000
04f9000efa0002100003fe0002c00004f90011fa0002200006fe0005600002023198fc0014fd000504633020000cfe0005300002064a58fc0014fd00050c94b0600008fe0005100003020a40fc0014fd0005041480400010fe0005080001021240fc0014fd0005042480400020fe0005040001022240fc0014fd0005e44480
800020fe0005040000824a40fc0014fd0005049480800040fe0005020000877980fc0012fd00050ef300800040fe000302000080fa000ffa0002800080fe000301000080fa000ffa0002800080fe000301000080fa0010fb000301000080fe000301000040fa0010fb000301000080fe000301000040fa000efb0002010001
fc0002800040fa0010fb000a0100010000100000800040fa0010fb000a0100010000300000800040fa0010fb000a0100010000100000800040fa0010fb000a0100010000100000800040fa0010fb000a0100010000100000800040fa0010fb000a0100010000100000800040fa0010fb000a0100010000380000800040fa00
0efb0002010001fc0002800040fa0010fb000301000080fe000301000040fa000ffa0002800080fe000301000080fa000ffa0002800080fe000301000080fa000ffa0002800080fe000301000080fa000ffa0002800040fe000302000080fa000ffa0002800040fe000302000080fa000efa0002400020fe0002040001f900
0efa0002400020fe0002040001f9000efa0002600010fe0002080003f9000efa0002200008fe0002100002f9000efa000220000cfe0002300002f90098001800f500680131012000f5006c0131012000f5006c0131012000000efa0002100006fe0002640004f9000efa0002100003fe0002c20004f9000efa000808001180
0001810008f9000efa0008080020600006008008f9000efa000804004038001c004010f9000efa00080200800f00f0002120f9000efa000803210000ff00001160f9000cfa00010122fc00010940f9000bf90000a4fc00010580f9000bf9000068fc000103f0f9000bfa000103f0fc000001f8000af9000020fb000080f900
0af9000040fb000040f9000af9000080fb000020f9000afa000001fa000010f9000afa000002fa000008f9000afa000004fa000006f9000afa000008fa000001f9000afa000010f9000080fa000afa000020f9000040fa000afa000040f9000020fa000afa000080f9000010fa000afb000001f8000008fa000afb000002f8
000004fa000afb000004f8000002fa000afb000008f8000001fa000afb000010f7000080fb000afb000020f7000040fb000afb000040f7000020fb000afb000080f7000010fb000afc000001f6000008fb000afc000002f6000004fb000cfc000004f60002030fe0fd000dfe000203f808f500013018fd000dfe00020c0630
f50001c006fd000efe0002300180f60002018003fd000ffe00026000c0f6000303000180fe000ffe0002c00060f6000302000080fe000ffe0002800020f6000304000040fe001005000001000010f6000304000040fe001005000001018010f6000308008020fe001005000002024008f6000308018020fe00100500000200
4008f6000308028020fe001005000002018008f6000308048020fe001005000002004008f6000308078020fe001005000002024008f6000308008020fe001005000002018008f6000308008020fe001005000002000008f6000304000040fe001005000001000010f6000304000040fe001005000001000010f60003020000
80fe000ffe0002800020f6000303000180fe000efe0002c00060f60002018003fd000dfe00026000c0f50001c006fd000dfe0002300180f500013018fd000cfe00010c06f400010fe0fd0007fe000103f8ee0002e90002e90002e90002e90098001801310068014701200131006c014701200131006c01470120000002e900
02e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e900a0008fa00083ff}}\par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Fig. 4.} Projection along C\endash
Cl bond of Figure 1. Atom 2(C) is obscured behind atom 1(Cl). Atoms 6, 7, and 8 (joined to 5) are not shown. Dihedral angles for atoms 3 and 4 are indicated., and where a carbon (number 2) is obscured behind the chlorine atom (number 1).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960
From Figure 4, one can determine the dihedral angles to be input for the first five atoms. For atom number 1 (chlorine), there are no previously entered atoms and so the bond length, bond angle and dihedral angle have no physical meaning. We enter a dummy
index of zero, and the bond length, bond angle and dihedral angle are arbitrarily set to zero. For the second atom (carbon), the bond length will be 1.77\'81, i.e., the distance betwe
en carbon (atom 2) and chlorine (atom 1). The bond angle and dihedral angle have no physical meaning, and are arbitrarily set to zero. The third atom is one of the hydrogens. The indices in Table 1 indicate that this is bonded to atom 2 (carbon). The bond
angle is with respect to atoms 2 and 1, and is taken as the standard tetrahedral bond angle of 109\'a1
. At this stage, we are at liberty to define the plane with respect to which the dihedral angles for atoms 3, 4, and 5 will be given. In the present example,
we make use of the symmetry of the molecule by taking this plane (which by definition includes atoms \ldblquote 0\rdblquote , 1, and 2) to include atom 5. The dihedral angle for atom 3 is then the angle required to rotate the (\ldblquote 0\rdblquote
,1,2,5) reference plane onto a plane containing atoms 1, 2, and 3. This dihedral angle is \endash 120\'a1
. Atom 4 follows in a similar vein. The indices indicate that it is bonded to atom 2. The bond angle is with respect to atoms 2 and 1, and is again 109\'a1. The dihedral angle is 120\'a1. Atom 5 is the other carb
on atom. The indices indicate that it is bonded to atom 2. The bond angle is with respect to atoms 2 and 1, and is 109\'a1
. Since this carbon lies in the plane which we have defined as a reference for the dihedral angles, its dihedral angle is zero. For atoms
6, 7, and 8, the reference plane for the dihedral angles contains the next three atoms above these in the branching diagram, i.e., atoms 1, 2, and 5. Consider atom 6. The dihedral angle is the angle required to rotate the (1,2,5) plane onto a plane contai
ning atoms 2, 5, and 6. Since atoms 2, and 5 are common to both planes, we look along the 2\endash 5 bond and determine the angle of rotation required to superimpose the two planes. This is shown in Figure 5.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \tab {{\pict\macpict\picw154\pich173
07ec007e0082012b011c1101a00082a0008e01000a0000000002d00240980014007e008000ba0120007e008200ba011c007e008200ba011c000002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0007f800017fc0f80006f8000080f70008f90002080006f80008f90002100001f80009f90003600000c0f90009f9
0003c0000060f9000afa00040180000030f90002ed0002ed000afa000004fe000004f9000afa000004fe000004f9000afa000008fe000002f9000afa000008fe000002f9000afa000008fe000002f9000afa000010fe000001f9000afa00041000040001f90008fa000210000cf70006f8000004f70006f8000004f70008f8
0002040001f9000afa00041000040001f9000afa000410000e0001f9000afa000008fe000002f9000afa000008fe000002f9000afa000008fe000002f9000afa000004fe000004f90006f6000004f90006f6000008f90006fa000001f50007fa00010180f60009f90003c0000060f90009f90003600000c0f90008f9000210
0001f80006f7000006f8000bf90005018018000030fb000cfa00020c007ffe000048fb000cfa0002120004fe000048fb000afa000012fc000030fb0011fe000403f000000cfd0003630003e0fd0012fe000c0c0c0018c00004000084800c18fd0012fe000c300300212000040008e4803006fd0012fe000c60018039200004
000894806003fd0012fe000c40008025200004003e94804001fd0013fe0004800041e520fe0005089480800080fe0013fe000d800040252001ffc0086300800080fe00141000000101e02018c03e003e000001018040fe0014100000010120200001c00001800001024040fe001410000001002020000f0004007000010240
40fe00141000000100402000180004000c0001018040fe0014100000010040200020000400020001024040fe00141000000100802000c0000400018001024040fe0013fe000d8080400100000400004001018040fe0098001400ba008000f6012000ba008200f6011c00ba008200f6011c000011fe000380004002fc000420
00800080fe0011fe00034000800cfc00041800800080fe0010fe000360018018fc00030c004001fd0012fe000c30030030000004000006006003fd0012fe000c0c0e002000000400000200b006fd0012fe000c03f10040000004000001410c18fd0010fc000a82800000040000014203e0fd000efc00046280000004fe0000
c4fb000bfc000013fb000103c8fb000bfc00010fc0fb000030fb000afc000006fa000040fb000efc000001fe000407e0000080fb000cfb0006c00000041c0001fa000cfb000620000004030002fa000cfb00061000000400c004fa000cfb00060c000004003018fa000cfb00060200007f801820fa000cfb00060100018060
0c40fa000bfa0005c00600180280fa000afa00042008000403f9000afa00041010000203f9000bfa00050c2000010c80fa000bfa00050260000190c0fa000bfa000501c00000a040fa000cf9000680000040600003fc000df900078000004020000480fd000df900078000004020000480fd000dfa000701000c0020200003
fc000dfa00070100120020108c60fc000dfa00070100020020119290fc000dfa00070100040020109290fc000dfa00070100080020108c90fc000dfa00070100120020109290fc000dfa000701001e0020109290fc000cf900068000004011cc60fc000af900048000004010fa000af900048000004020fa000af900044000
008020fa000af900046000018020fa000af900042000010060fa000af900041000020040fa000af9000408000400c0fa000af900040600180080fa0009f9000301806001f90008f800027f8003f90008f80002080002f90008f8000208000cf90008f80002080018f90008f80002080030f90008f800020900c0f90007f800
010a03f80007f800010c1cf80007f800010fe0f80006f800000cf70006f800000af70006f8000009f7000bfe000109c0fd000008f7000bfe00013030fd000008f7000ffe000040fc000008fd00010138fd000ffe000080fc000008fd00010606fd0098001400f60080012b012000f60082012b011c00f60082012b011c0000
0efd000002fd000008fd000008fc000efd000001fd000008fd000010fc0010040000020001fd000008fb000040fe001105000004030080fe000008fb000020fe001305000004048080fe000008fd0002400020fe0012040000040080fd000008fd0002800010fe0010fe000003fc000008fd0002802010fe0010fd00018080
fe000008fd00018060fd001105000004048080fe000008fc0000a0fd001305000004030080fe000008fd0002012010fe0012040000020001fd000008fd000281e010fe0012040000020001fd000008fd0002802010fe000cf8000008fd0002402020fe000cf8000008fd0002400020fe000bfe00014008fd00007ef7000dfe
00013030fe0002018180f80012fe00010cc0fe0002060060fe00010801fd000df90002080030fe00010606fd000df90002180018fe00010138fd0008f90002100008f80008f9000230180cf80008f90002202004f80008f90002203804f80008f90002202404f80008f90002202404f80008f90002202404f80008f9000230
180cf80008f90002100008f80008f90002180018f80008f900020c0030f80008f90002060060f80008f90002018180f80006f800007ef70002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed0002ed00a0008fa00083ff}}\par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Fig. 5.} Projection along C\endash
C bond of Figure 1. Atoms 1, 3, and 4 are in the foreground. Atom 5 is positioned behind atom 2. The plane containing atoms 1, 2, and 5, indicated by the dashed line, is used to define the dihedral angles for atoms 6, 7, and 8.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 The dihedral angle for atom 6 is 180\'a1. For atoms 7 and 8, the dihedral angles are \-\endash 60\'a1 and +60\'a1
respectively. The bond angles for the final three atoms (with respect to atoms 5 and 2 in each case) are all 109\'a1. This completes the specification of the structure of the molecule.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960
One point which may be apparent is that if one is to use the exact bond lengths and bond angles, then it is rather difficult to estimate the exact dihedral angles. For this reason, it is simpler to use standard bond angles (e.g., 109\'a1
for tetrahedrally bonded carbons, 120\'a1 for doubly bonded carbons, etc.). For most purposes, the moments of inertia are sufficiently accurate.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
{\b Chloroethane transition state\par
}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 We determine the moments of inertia for the four\endash centre transition state for the elimination of HCl from chloroethane by specifying the bond lengths for the carbon\endash carbon, carbon\endash
hydrogen (four ordinary and one stretched), and carbon\endash chlorine bonds, and by assuming approximate bond angles as follows. The atoms in the transition state are numbered as in Figure 6.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \tab {{\pict\macpict\picw351\pich124
035600000016007c0175110101000a0000000002d0021c08000a71001e004500300053003d0045003c0045003d00530033005100300045003c220047007d000022004700800000220048008300ff220048008600ff220048008900ff2200450049040023000022004500510400230000220045005904002300002200450061
040023000022004500690400230000220045007100002200400049280071001e0047007b005500830047007b0047007c00550083005500800047007b22004500390000220045003700002200450035000022004500320001220045003000012200250040000423000022002d00400004230000220035004000042300002200
3d0040000022001a007cff04230000220021007bff04230000220029007aff042300002200310079ff0423000022003900780002220025004003ff230000220023004704ff230000220022004f04ff2300002200200057040023000022001f005f04ff23000022001e006704ff23000022001c006f04ff23000022001b0077
04ff23000071001e004300da005500e7004300e6004300e7005500dd005300da004300e622004300e3000022004300e0000022004300dd010122004300db000122004300d8000222004100f2040023000022004100fa040023000022004101020400230000220041010a04002300002200410112040023000022003c00f224
00220039012000fc230000220031012001fc230000220029012101fc230000220021012201fc230000220019012301fe71001e0043012200560134004301220043012300540134005601310043012222004301250100220044012901ff220044012d01ff220045013101fe220045013501fe2200170124fc00230000220017
011cfc012300002200180114fc01230000220019010cfc0023000022001a0104fc0023000022001b00fcfc0023000022001b00f4fe0122002200e8000423000022002a00e8000423000022003100e9000423000022003900e901020300140d000a2be74203283529280047003f014328005a002c014828004b002601482800
24003d01482b38220143280019007802436c2b0a45014828004e009001482800160122032831292b11460328332928004701390328342928004201190328322928005c00d70328362928002100e40328382928004700cb03283729ff}}\par
\par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Fig. 6.} Sample numbering scheme for the transition state in the chloroethane elimination reaction.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 This numbering scheme is of course not unique. Following the procedure used above for the chloroethane molecule, approximate bond lengths, bond angles and dihedral angles assigned as in Table 2.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Table 2.} Bond angles, dihedral angles and bond lengths for specification of the structure of the transition state in the chloroethane elimination reaction.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard\plain \s6\qj\li720\ri331\sl280\brdrb\brdrs \brdrbtw\brdrs \tx720\tqr\tx1260\tqc\tx2250\tqr\tx3960\tqc\tx4860\tqc\tx5760\tqc\tx7020 \f20\fs18 \par
____________________________________________________________________\par
\tab Index for\tab Index for\tab Bond angle\tab Dihedral\tab Bond\tab Mass of\par
\tab present atom\tab attached atom\tab \tab angle\tab length\tab present atom\par
\par
\tab 1\tab 0\tab 0\tab 0\tab 0\tab 35.45\par
\tab 2\tab 1\tab 0\tab 0\tab 1.84\tab 12.0\par
\tab 3\tab 2\tab 100\tab \endash 120\tab 1.09\tab 1.0\par
\tab 4\tab 2\tab 100\tab +120\tab 1.09\tab 1.0\par
\tab 5 \tab 2\tab 100\tab 0\tab 1.36\tab 12.0\par
\tab 6\tab 5\tab 115\tab \endash 105\tab 1.09\tab 1.0\par
\tab 7\tab 5\tab 115\tab +105\tab 1.09\tab 1.0\par
\tab 8\tab 5\tab 100\tab 0\tab 1.6\tab 1.0\par
____________________________________________________________________\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 The bond angles and dihedral angles were obtained by estimating values intermediate between those of the reactant (chloroethane) and the products (HCl above planar ethene).\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 The dihedral angles for atoms 3, 4, and 5 may be determined by looking along the 1\endash 2 (i.e., Cl\endash C) bond. From Figure 4, we can see that these angles are \endash 120\'a1, +120\'a1 and 0\'a1
respectively for the reactant. However, if one imagines HCl above planar ethene, a projection along the corresponding Cl\endash C axis will again give dihedral angles of \-\endash 120\'a1, +120\'a1 and 0\'a1
for atoms 3, 4 and 5 (assuming the Cl is aligned directly above a carbon. We have therefore assumed in Table 2 that the dihedral angles of atoms 3, 4 and 5 take the values of \endash 120\'a1, +120\'a1 and 0\'a1
respectively in the transition state also. The dihedral angles for atoms 6, 7 and 8 are in each case defined relative to a plane containing atoms 1, 2 and 5 (Figure 5). Since we are assuming
a planar ring for the transition state, atom 8 (the reacting hydrogen) is also part of this plane (Figure 7).\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \tab \tab {{\pict\macpict\picw178\pich169
07a8007b0061012401131101a00082a0008e01000a0000000002d00240980018007b006000b70118007b006100b70113007b006100b70113000002e90002e90002e90007f70001063ef50007f700013c03f50008f70002c00010f60006f5000008f60006f5000006f60009f800030c000002f60006f8000010f3000af80000
30fe000040f7000af8000060fe000060f7000af8000040fe000020f7000af8000080fe000010f70006f4000010f70006f4000008f70002e90006f9000001f20006f9000002f2000bf90005020000400004f7000bf90005020000c00004f7000bf90005020000400004f7000bf90005020000400004f70008f60002400004f7
0008f60002400004f70008f60002e00004f7000af9000002fd000004f70006f9000001f20006f9000001f20006f9000001f2000af8000080fe000010f7000af8000080fe000010f7000af8000040fe000020f70006f4000060f7000af8000010fe0000c0f7000af8000010fe000080f70006f800000cf30009f80003060000
04f60009f8000301000008f60008f70002c00030f60007f6000101c0f60007f7000107f8f50002e90002e90002e90007f7000103f0f50007f700010c0cf50007f700011002f50007f700012001f50008f70002400080f60008f70002800040f60008f70002800040f60009f800030100c020f60009f8000301012020f60009
f8000301012020f60009f800030100c020f60009f8000301012020f60009f8000301012020f60009f800030100c020f60008f70002800040f60098001800b7006000f3011800b7006100f3011300b7006100f30113000008f70002800040f60008f70002400080f60007f700012001f50007f700011002f50007f700010c0c
f50007f7000103f0f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50006f6000040f50007f700011ffff50009f8000301e040f0f60009
f800030e00400ef6000af800043000400180f7000af80004c000400060f7000bf90005030000400018f7000bf90005040000400004f7000bf90005180000400003f7000cf9000620000040000080f8000ff90009400000400000400233d8fb000ff9000980000040000020064a18fb0010fa000001fe000640000010024b80
fb0013fd00030119ec02fe000640000008024840fb0013fd000303250c04fe000640000004024840fb0013fd00030125c008fe000640000002024a40fb0013fd000301242018fe000640000003073180fb0010fd000339242010fe000340000001f80011fd000301252020fe000040fe000080f90011fd00030398c040fe00
0040fe000040f9000efa000040fe000040fe000040f9000efa000080fe000040fe000020f9000efa000080fe000040fe000020f9000efb000001fd000040fe000010f9000ffb000001fe000107fcfe000010f9000ffb000003fe00011c07fe000018f9000ffb000002fe00056000c0000008f9000ffb000902000001800030
000008f9000ffb000904000003000018000004f9000ffb00090400000600000c000004f9000ffb000904000004000004000004f9000ffb000904000008000002000004f9000ffb000904000010000001000004f9000ffb000908000010000001000002f9000ffb000908000010000001000002f9000ffb0009080000200060
00800002f9000ffb000908000020009000800002f9000ffb000908000020001000800002f9000ffb000908000020002000800002f9000ffb000908000020004000800002f9000ffb000908000020009000800002f90098001800f300600124011800f300610124011300f300610124011300000ffb00090800002000f00080
0002f9000ffb000908000010000001000002f9000ffb000908000010000001000002f9000ffb000908000010000001000002f9000ffb000904000008000002c00004f9000ffb000904000064000004380004f9000ffb00090400038600000c070004f90010fb000a04001c0300001800e02480fa000ffb00092480e0018000
30001c15f9000ffb0009150700006000c000038ef9000ffb00050e3800001c07fe000074f9000ffb000505c0000007fcfe00000ef9001206000003f000001ef9000401c00003f0fd00110600000c0c0000e0f8000338000c0cfd001005000010020007f7000307001002fd000f05000020010038f60002e02001fd00100500
00400081c0f600031c400080fe000f04000080004ef5000303800040fe000e040000800070f40002800040fe000f04000100c020f500030101e020fe000f040001010020f5000301012020fe000f04000101c020f5000301002020fe000f040001012020f5000301004020fe000f040001012020f5000301004020fe000f04
0001012020f5000301008020fe000f04000100c020f5000301008020fe000e040000800040f40002800040fe000e040000800040f40002800040fe000e040000400080f40002400080fe000c0300002001f300012001fd000c0300001002f300011002fd000c0300000c0cf300010c0cfd000c03000003f0f3000103f0fd00
02e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e90002e900a0008fa00083ff}}\par
\par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Fig. 7.} Projection along C\endash C (2\endash 5) bond of Figure 6, detailing dihedral angles for atoms 6, 7 and 8 (non-reacting H\rquote s and the reacting H res
pectively) in the transition state. Atom 5 is obscured behind atom 2. Atom 1 (Cl) is in the foreground. The reference plane includes atoms 1, 2, 5 and 8. For simplicity, atoms 3 and 4 (non-reacting H's) are not included in the foreground.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960
The dihedral angle for atom 8 is therefore zero. With the assumption of a symmetric transition state, atoms 6 and 7 will have dihedral angles of opposite sign but the same magnitude. The dihedral angles would change from 120\'a1
for an eclipsed configuration of chloroethane to 90\'a1 for HCl above planar ethene.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b Methyl radical}{\b\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\b\v recombination (association) rate coefficient}}}{\b transition state\par
}\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Table 3.} Bond angles (degrees), dihedral angles (degrees) and bond lengths (\'81
) for specifying structure of loose transition state in methyl radical reaction (sample fragment separation of 4.5 \'81 used).\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard\plain \s6\qj\li720\ri331\sl280\brdrb\brdrs \brdrbtw\brdrs \tx720\tqr\tx1260\tqc\tx2250\tqr\tx3960\tqc\tx4860\tqc\tx5760\tqc\tx7020 \f20\fs18 \tab Index for\tab Index for\tab Bond angle\tab Dihedral\tab Bond\tab Mass of\par
\tab present atom\tab attached atom\tab \tab angle\tab length\tab present atom\par
\par
\pard\plain \s12\qj\li720\ri331\sl280\brdrb\brdrs \brdrbtw\brdrs \tx720\tqr\tx1260\tqr\tx2520\tqr\tx3960\tqr\tx5220\tqr\tx6030\tqr\tx7200 \f20\fs18 \par
\tab 1\tab 0\tab 0\tab 0\tab 0\tab 1.0\par
\tab 2\tab 1\tab 0\tab 0\tab 1.09\tab 12.0\par
\tab 3\tab 2\tab 120\tab \endash 90\tab 1.09\tab 1.0\par
\tab 4\tab 2\tab 120\tab 90\tab 1.09\tab 1.0\par
\tab 5\tab 2\tab 90\tab 0\tab 4.5\tab 12.0\par
\tab 6\tab 5\tab 90\tab 180\tab 1.09\tab 1.0\par
\tab 7\tab 5\tab 90\tab \endash 60\tab 1.09\tab 1.0\par
\tab 8\tab 5\tab 90\tab 60\tab 1.09\tab 1.0\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
The specification of bond angles and dihedral angles requires some explanation. The numbering scheme is indicated in Figure 8.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {{\pict\macpict\picw397\pich134
029c0006000a008c0197110101000a0000000002d0021c08000a22003d002f040023000022003d0037040023000022003d003f040023000022003d0047040023000022003d004f040023000022003d0057040023000022003d005f040023000022003d0067040023000022003d006f040023000022003d0077040023000022
003d007f040023000022003d0087000071001e00420084005800900042008f0042009000580087005600840042008f220045009101ff220048009401ff22004c009602fe22004f009902fe220052009c02fe220035008f00ea71001e0024001a0037002700370026003700270024001d0026001a0037002622003200280001
22002e002a010122002a002d0101220026002f0201220022003202012200420026001522003d015dfc0023000022003d0155fc0023000022003d014dfc0023000022003d0145fc0023000022003d013dfc0023000022003d0135fc0023000022003d012dfc0023000022003d0125fc0023000022003d011dfc002300002200
3d0115fc0023000022003d010dfe0071001e004501570058016500450164004501650058015a00560157004501642200470166000022004a016900ff22004d016c00ff220050016e01fe220053017101fe220035016400ea2200450101001071001e002500f5003501020035010100350102002500f8002700f50035010122
00310103000022002d0105010022002901070201220025010a0101220021010c02010300140d000a2bfb5e03283629280040002601432965014328001c008b01482800630080014828005c00a00148280063002501482800230017014828001d003301482bc923032835292b65010328322928001c015f0328312928006101
530328332928005e01710328342928001f00ec0328372928001d010903283829ff}}\par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Fig. 8.} Numbering scheme, for loose transition state in methyl-methyl reaction, used to obtain bond and dihedral angles of Table 3.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \tab The bond angle for atom 3 is relative to atoms 1 and 2 and is 120\'a1, since the fragment is assumed to be planar. The same holds for the bond angle for atom 4. We have c
hosen the reference plane for the dihedral angles of atoms 3, 4 and 5 so that it includes atom 5, as indicated in Figure 9.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \tab \tab {{\pict\macpict\picw172\pich185
0745007f008e0138013a1101a00082a0008e01000a0000000002d00240980018007f008800bb0140007f008e00bb013a007f008e00bb013a000002e90002e90002e90002e90002e90002e90007f60001ff80f60008f700020380e0f60008f700020c0018f60008f70002100004f60008f70002600003f60009f70003c00001
80f70009f7000380000080f7000af8000001fe000040f7000af8000001fe000040f7000af8000002fe000020f7000af8000002fe000020f7000af8000004fe000010f7000af8000004fe000010f7000af8000404003c0010f7000af800040400200010f7000af800040400380010f7000af800040400040010f7000af80004
0400040010f7000af800040400240010f7000af800040200180020f7000af8000002fe000020f7000af8000001fe000040f7000af8000001fe000040f70009f7000380000080f70009f70003c0000180f70008f70002600003f60008f70002100004f60008f700020c0018f60008f700020380e0f60007f60001ff80f60002
e90002e90006f6000008f50006f6000008f50006f6000008f50006f6000008f50002e90002e90002e90006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50002e90002e90006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50002e90002e90002e900
98001800bb008800f7014000bb008e00f7013a00bb008e00f7013a000006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50002e90002e90002e90006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50002e90002e90002e90006f6000008f50006f600
0008f50006f6000008f50006f6000008f50006f6000008f50002e90002e90002e90006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50002e90002e90002e90006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50002e90002e90002e90006f6000008
f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50002e90002e90002e90006f6000008f50006f6000008f50006f6000008f50006f6000008f50006f6000008f50002e90002e90002e90006f6000008f50006f6000008f5000afa0004c660000008f50010fb000a0129600001ffc00000c660fa009800
1800f700880133014000f7008e0133013a00f7008e0133013a000010fb000a012900003e083e00012960fa000ffb000901290000e00003800129f9000ffb00030ee90007fe0002700129f9000efa000229001cfe00021c00e9f9000efa0008c60030000800060029f9000cf80006c00008000180c6f9000cf9000601800008
0000c0f8000cf9000602000008000020f8000cf9000604000008000010f8000cf9000608000008000008f8000cf9000610000008000004f8000af9000020fc000002f8000af9000040fc000001f8000af9000080fb000080f9000efa00040180000008fe0000c0f9000efa000001fe000008fe000040f9000efa000002fe00
0008fe000020f9000efa000002fe000008fe000020f9000efa000004fe000008fe000010f9000afa000004fa000010f9000afa000008fa000008f9000efa000008fe0004ff80000008f9000efa00081000000380e0000004f9000efa00081000000c0018000004f9000efa0008100000100004000004f90015fe000f01fc00
00200000607f0300000200007ffc0016fe001006030000200000c180c1800002000180c0fd0016fe00100800800020000082002080000200020020fd0016fe00101000400020000104001040000200040010fd0016fe00102000200040000108000840000100080008fd0016fe001040001000400002100004200001001000
04fd0016fe00104000100248000210000420000920100004fd0016fe00108060080150000420080210000540200402fd0016fe001080900800e0000420180210000380200c02fd0016fe00108010080040000420080210000100201402fd0016fe000280600bfeff04f420080213feff02a02402fd0016fe0002801008fe00
040420080210fe0002203c02fd0016fe0002809008fe00040420080210fe0002200402fd0016fe0002806008fe000404201c0210fe0002200402fd0016fe0002400010fe00040410000410fe0002100004fd0016fe0002400010fe00040210000420fe0002100004fd0016fe0002200020fe00040208000820fe0002080008
fd0016fe0002100040fe00040104001040fe0002040010fd0016fe0002080080fe00040102002040fe0002020020fd0014fe00010603fc00038180c080fe00020180c0fd0012fe000101fcfc0003c07f0180fd00007ffc0008f70002600003f60008f70002100004f60008f700020c0018f60008f700020380e0f60007f600
01ff80f60002e90002e90002e90002e90002e90002e90002e90002e90002e90098001801330088013801400133008e0138013a0133008e0138013a000002e90002e90002e90002e90002e900a0008fa00083ff}}\par
\par
\pard\plain \s9\qj\ri360\sl240\tx720\tqc\tx3960\tqr\tx7920 \f20\fs18 {\b Fig. 9.} Projection along (1,2) bond of Figure 8 indicating dihedral angles for atoms 3 (H), 4 (H), and 5(C). Atoms 6, 7, and 8 (attached to 5) are not included.\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 The dihedral angle for atom 3 is therefore the angle required to rotate the plane containing atoms 1, 2 and 5 onto a plane containing atoms 1, 2, and 3. From Figure 9, it is apparent that this angle is \endash 90\'a1
. The dihedral angle four atom 4 follows similarly as +90\'a1. Since atom 5 lies in the plane, its dihedral angle is 0\'a1. The bond angle for atom 5, relative to atoms 1 and 2, can be seen to be 90\'a1
. The bond angles for atoms 6, 7, and 8 in Table 3 are relative to atoms 2 an
d 5 in each case (since these are the next two atoms above them in a branching diagram such as Figure 3). Since we are assuming planar methyl fragments in the loose transition state, these bond angles are also 90\'a1
. The dihedral angles for these atoms, expressed relative to the plane containing atoms 1, 2 and 5, are the same as for chloroethane (i.e., 180\'a1, \endash 60\'a1 and +60\'a1 respectively).\par
\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 \page {\v {\tc\tcl1\pard\plain \s254\ri29\sb120\sl280\keepn\tx720\tqc\tx3960 \b\f20 \tab \tab NOTES}}\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b Note A} {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v interconversion, energy}}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v energy interconversion}}}
The program automatically calculates (by numerical integration) the values of <{\f23 D}{\i E}>, <{\f23 D}{\i E}{\fs18\up6 2}>{\fs18 \|S\|UP4(1)\|S\|UP3(/)\|S\|UP2(2)}, <\'c6{\i E}{\fs18\dn4 down}> and the collision efficiency {\i\f23 b}
. The user can thus easily interconvert between whatever measure of {\i P}({\i E,E'}\|D\|FO1()) is required, desired and/or available (note that inaccuracies inevitable in the numerical integration will mean that the values of <\'c6{\i E}{\fs18\dn4 down}
>, etc., printed by the program will be slightly different from their exact, i.e., analytical, values). Note also that all these measures of {\i P}({\i E,E'}\|D\|
FO1()) may depend on temperature and the internal energy. The program includes a cutoff parameter (ERR3) which truncates {\i P}({\i E,E'}\|D\|
FO1()); this makes the matrix corresponding to eq 5 banded, and the band width NBAND is printed out by the program. The default choice of {\i P}({\i E,E'}\|D\|FO1()) is th
e exponential model; in addition, the program permits very flexible functional forms to be chosen.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \tab If one wishes to interconvert <{\f23 D}{\i E}>and <{\f23 D}{\i E}{\fs18\up6 2}>{\fs18 \|S\|UP4(1)\|S\|UP3(/)\|S\|UP2(2)}
without doing anything else, this can be achieved by using RRKM and feeding in the reactant frequencies as those of the molecule where one wants do to the interconversion, as dummy frequencies for the transition state (easily done by simply feeding in one
less frequency than the \ldblquote reactant\rdblquote ). The output on Unit 10 is then used directly in MASTER, and the interconversion found in the output listing of <{\f23 D}{\i E}> and <{\f23 D}{\i E}{\fs18\up6 2}>{\fs18 \|S\|UP4(1)\|S\|UP3(/)\|S\|
UP2(2)} values as functions of internal energy.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b Note B} The program carries out the calculation for successively smaller values of the grain size ({\f23 d}{\i E}
) used in the finite difference equivalent of the integral eq 5. The first eigenanalysis is for a large grain size ({\f23 d}{\i E} = 200 cm{\fs18\up6 \endash 1}, say), and the resulting {\i g}({\i E}) is used to yield an interpolated {\i g}({\i E}
) at a smaller grain size. For informative purposes, the number of ite
rations required for convergence within the desired tolerance is printed, as well as the matrix size (total dimension) of the finite difference equivalent of eq 5. The program defaults to a final grain size of 100 cm{\fs18\up6 \endash 1}
, which is adequate for almost all purposes.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b Note C} {\i Treatment of Rotational Degrees of Freedom.}{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v rotations}}}
A proper treatment of angular momentum is particularly important in regard to falloff effects in reactions (such as {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v recombination (association) rate coefficient}}}recombination or ass
ociation reactions) wherein reactant and activated complex have very different moments of inertia. For convenience, certain salient points are summarized here. Note that program GEOM does all the necessary calculations for these treatments from input bond
lengths and angles.\par
(i) All information about a rotational degree of freedom is input to the RRKM program as the value of the rotational constant {\i B} (corresponding to its moment of inertia {\i I}\|D\|FO1()), its dimension (1, 2 or 3) and the quantity [symmetry num
ber]/[number of optical isomers]; see Gilbert and Smith). Note that if {\i B} is in cm{\fs18\up6 \endash 1} (as required by the program) and {\i I} is in (a.m.u. \'81{\fs18\up6 2}), then {\i B} = 16.86/{\i I}. {\v {\xe\pard\plain
\qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v moment of inertia conversion factor}}rotational constant conversion factor;}\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 (ii) Internal rotations (such as one might use to approximate the motion of a methyl group) may be either free or hindered. If a rotor with moment of inertia {\i I}
is hindered so that it can only rotate about an angle 2{\i\f23 q} then it behaves like a free rotor with an effective moment of inertia given by {\fs16\up4 1}{\fs16 /}{\fs16\dn4 2}(1\endash cos{\i\f23 q}){\i I} instead of {\i I}
. One inputs hindered rotors into the RRKM program by using this effective moment of inertia. Note: all internal rotations are active (i.e., their energy is available for crossing the barrier). All calculations (including calculation of the hindered {\i B
} value) are done using GEOM. See also Note H.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 (iii) The program also requires information about the overall (external) moments of inertia. If both the reactant molecule and the activated complex have identical moments of inertia (i.e., {\i I} = {\i I}{\fs18\up6
\'a0}) then the final results are independent of the value of {\i I}, and one can input dummy (but equal!) values of {\i I} and {\i I}{\fs18\up6 \'a0} (or rather, the corresponding values of {\i B} and {\i B}{\fs18\up6 \'a0}).\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 (iv) If the external moments of inertia of molecule and complex are not equal, appropriate values must be input. The three external moments of inertia (denoted {\i I}{\fs18\dn4 a}, {\i I}{\fs18\dn4 b}, {\i I}{
\fs18\dn4 c}) for the molecule, and {\i I}{\fs18\dn4 a}{\fs18\up6 \'a0} etc. for the complex) may, in the most elementary treatment (see next part of this note for a better treatment!) all be ass
umed to be inactive (i.e., unable to interchange energy with any of the internal degrees of freedom). In such a case, the values of {\i I} and {\i I}{\fs18\up6 \'a0} are {\i I} = ({\i I}{\fs18\dn4 a}\~{\i I}{\fs18\dn4 b}\~{\i I}{\fs18\dn4 c})\|S\|UP4({
\fs18 1)\|S\|UP3(/)\|S\|UP2(3}), and similarly for the complex.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 (v) {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v symmetry number}}}
However, since one external rotational degree of freedom (corresponding to the K rotational quantum number) is not conserved and its energy is available for crossing the barrier, this degree of freedom should be treated as an active rotor; the other two ar
e inactive. Now, one generally finds for a typical molecule that two moments of inertia are equal or nearly equal ({\i I}{\fs18\dn4 a} {\f23 \'bb} {\i I}{\fs18\dn4 b}), say \par
(symmetry often results in an exact equality) while the third ({\i I}{\fs18\dn4 c}) is different. It is {\i I}{\fs18\dn4 c} that corresponds to the {\i K}
quantum number, i.e., to the active external rotation. This is indicated to the program by feeding in data corresponding to a two-dimensional external rotor with {\i I} = ({\i I}{\fs18\dn4 a} {\i I}{\fs18\dn4 b})\|S\|UP4({\fs18 \|S\|UP4(1)\|S\|UP3(/)\|S
\|UP2(2)}) and a one-dimensional free internal rotor with {\i I} = {\i I}{\fs18\dn4 c}. It is essential
to note that in order to have the same overall thermodynamics, whether rotors are treated as active or inactive (clearly a necessary requirement for physical consistency), care must be taken with the symmetry numbers. If for convenience one takes the symme
try number of the two-dimensional external inactive rotation to be the same as that of the overall molecule, then the symmetry number of the one-dimensional free internal rotation corresponding to {\i I}{\fs18\dn4 c}
must be taken as 1. (Note that the rate coefficient is only proportional to ratios of symmetry numbers, so that one can of course have alternative ways of assigning these symmetry numbers from that just given). Note that in the case of radical {\v {\xe
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 recombination{\v (association) rate coefficient}}} reactions any {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 electronic degeneracy}}
effect may be taken into account through the symmetry numbers; this is discussed extensively in Gilbert and Smith. This can be accommodated by changing the {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 reaction path degeneracy}}
. Alternatively, this effect may be taken into account when calculating the equilibrium constant via the electronic partition function.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960
(vi) For simplicity of use, the default option in the program is that all external rotations are treated as inactive, and there is no provision for direct input of the dimensionality of the external rotations (which is 3 under the default). If (as is advis
able for physical accuracy) the user wishes to make one external rotational degree of freedom (of both molecule and complex!) active, this is indicated by inputting negative external {\i B} values. The program then takes these to be (positive) {\i B}
values corresponding to two-dimensional external degrees of freedom; {\i I}{\fs18\dn4 c} must then be input as the corresponding {\i B} value for an active rotor, with symmetry number {\i\f23 s}
= 1, and dimension = 1. The program checks for the presence of the one-dimensional active rotors if this option is employed. Note that division into active and inactive rotors has no effect on the high-pressure rate, but has a strong effect in the falloff
regime.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 (vii) The program combines the inactive and active rotations in printing out the rotational partition functions.\par
(viii) Program GEOM takes input geometry and calculates moments of inertia, hindrances and other quantities required for the above analysis.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 (ix) {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v angular momentum conservation}}{\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v J-conservation}}}{\i
Conservation of angular momentum.} As stated, the master equation should properly be written in terms of total angular momentum {\i J} as well as in total energy {\i E}, both of which are conserved in an isolated molecule. If {\i I}{\fs18\up6 \'a0}
is much greater than {\i I}, angular momentum conservation effects become very significant. This effect is important in the case of reactions proceeding by a simple-fission{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v
simple-fission transition state}}} transition state: e.g., radical {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v recombination (association) rate coefficient}}}reactions and many decompositions. The program enables one to take {\i J}
conservation into account as described in Gilbert and Smith. This option is used by having JAV nonzero in program RRKM; if JAV=0, then angular momentum con
servation in the falloff regime is not taken properly into account. If the J-conservation option is used, then some additional information is required on the potential and transition state, as follows. (a) For each reaction channel, the values of RCPL and
REQ for each channel are read in, these being the distance between the two A\'c9B moieties (for the process AB{\f23 \'ae}
A+B) corresponding to the transition state and to the A-B equilibrium distance, respectively. (b) For each rotational energy considered the centrifugal barrier along the reaction path is located as the maximum in the effective potential: {\i V}{\dn4 eff}=
{\i V}({\i r},{\i J}=0)+{\i B}({\i r}){\i J}({\i J}+1) - where a symmetric rotor geometry has been assumed. In the case where the pivot atoms on each end of the breaking bond represent the centres of mass of the fragments then {\i I}({\i r}) = {\i\f23 m}{
\i r}{\up6 2}, where {\f23 m} represents the reduced mass of fragments A and B (the reduced mass is in fact calculated from the values BCMPLX and RCPL rather than entered directly). If this is not the case, and the centre of mass of at
least one fragment does not coincide with the fragment's pivot atom, then values of the rotational constant as a function of r must be entered. This option is specified by setting the molecular mass, WT1{\f23 <}
0. The program performs a linear interpolation of the entered points in subroutine BESTFIT, calculating a correlation coefficient which is output. If the correlation coefficient is less than 0.9 the program will abort. The location of the centrifugal barri
er is then calculated from the interpolated line of best fit. Note that if the pivot atom is almost the centre of mass of the fragment, as is the case with the O-H fragment in the reaction CH{\dn4 3}OH {\f23 \'ae} CH{\dn4 3}
+ OH, then entering a range of rotational constants along the reaction path is unlikely to cause more than 1 or 2% difference in any calculated rate coefficients. (c) Next, some information about the A\'c9
B interaction is read in (needed to locate the centrifugal barrier). The method assumes that this interaction is a Morse potential, and (given the dissociation energy for the process, {\i E}{\fs14\dn4 0}
) the program finds a best-fit Morse to input potential points. For each channel, one inputs (in the RRKM input file) the values of NV (the number of points to be input) and then NV values of RVCH and VCH, which are respectively the values of {\i r}
and of {\i V}({\i r}) to which the program will fit a Morse. It is essential to note that the J-average {\i k}({\i E}
) used in this solution depends on both temperature and pressure; thus if the J-conservation option is used, the RRKM program must be run for each temperature and pressure for which one requires a solution of the master equation. The calculation for a {
\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 multichannel reaction}} requires the low pressure limiting values of {\i k}{\fs18\dn4 uni} for each channel [see note C(ix)]. In order to obtain the correct {\i k}{\fs18\dn4 uni} \rquote
s, therefore, it is necessary to lower the pressure at which the {\i k}{\fs18\dn4 uni} are calculated until the ratio of {\i k}{\fs18\dn4 uni}\rquote
s approaches its constant, low-pressure limiting value. The values of RCPL, etc., specifying the Morse potential can be found, for example, using the Morse {\i\f23 b}\'e2 from the bond dissociation energy and vibrational frequency (Herzberg, \ldblquote
Spectra of Diatomics\rdblquote , p 101). If this J-conserving option is not used, then angular momentum conservation is taken into account only in the high pressure regime, by using a {\i k}({\i E}) multiplied by the factor {\i I}{\fs18\up6 \'a0}/{\i I}.
\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b Note D}
It is necessary to ensure that NN is chosen sufficiently large so that all significant values of the integrand in eq 7 are included. To check this, the program calculates the high-pressure frequency factor ({\i A}{\fs18\dn4 \'b0}
) analytically (from the transition-state thermodynamics) and numerically. If the two values are significantly different, a warning message is printed; however, see also Notes E and F. Note also that this check is not carried out if one uses the option to
account for {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 angular momentum conservation}}
in the falloff regime [Note C(ix)]; under these circumstances, the check for sufficiently high NN can be carried out by running the program with JAV=0.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b Note E} It is necessary to ensure that INC is chosen sufficiently small so that the integrals in eqs 5 and 7 are accurately approximated by the equivalent sum (100 cm{\fs18\up6 \endash 1}
is strongly recommended as being sufficient). As in Note D, this is checked through the numerical and thermodynamic values of {\i A}{\fs18\dn4 \'b0}
and of the partition function of reactant. If a warning message is printed, this is due to a combination of small inaccuracies in the semiclassical expressions for {\i\f23 r}({\i E}
) for internal rotors and numerical evaluation of the various integrals. These inaccuracies are automatically corrected as described in Note F. Since experience has shown that INC = 100 cm{\fs18\up6 \endash 1}
is always sufficiently small, MASTER assumes that an INC value of 100 cm{\fs18\up6 \endash 1} is used in calculating {\i k}({\i E}) and {\i\f23 r}({\i E}), and aborts if this is not so; however, the RRKM program may be run alone with INC \'ad 100.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b Note F} Another check on the accuracy of the numerical evaluation of eq 7 is through comparison of the value of {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}
) as found from numerical integration and from the transition state thermodynamics. One often finds that there is a difference of 10% or less between the exact and numerical values of {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}
) for molecules with moderately high frequencies and/or any molecule at sufficiently low temperatures (e.g., below 1000 K). As stated in Note E, the causes of this error are inaccuracies in the semiclassical form for {\i\f23 r}({\i E}
) for rotors and in the numerical evaluation of the denominator of eq 7 (which is equal to the internal partition function at high pressures). The RRKM program compares the two values of {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}
) and prints out a warning if the discrepancy exceeds 10% (this however is not done if the option is used to ensure the angular momentum conservation in the falloff regime). The high-pressure rate coefficient is then calculated by a different method (see
Appendix I) and an appropriate numerical correction factor determined. The numerical inaccuracy in this case is automatically corrected by multiplying all computed rate coefficients (including those fr
om the master equation) by the ratio of the numerically-evaluated {\fs28 \'ba}\|D\|FO1(){\i f}({\i E})\~d{\i E}
and the exact (internal) partition function. When the warning message is printed out, the RRKM program also prints out the appropriate correction factor, which is carried through into the master equation file for each input temperature and all rate coeffi
cients as printed are corrected by this factor. It also lists separately the corrected values of the various rate coefficients. If this numerical inaccuracy occurs, weak coll
ision contributions to the denominator of eq 7 will be negligible, and the same correction factor is thus also applicable to rate coefficients computed using MASTER. In many cases, this correction factor (CORRAT) is close to unity, and one does not need to
worry about it. CORRAT is always generated if the falloff J-conserving option is used.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b Note G} Any electronic degeneracy factors, which must be taken into account for recombination reactions, are included in the overall symmetry number. An extensive expos
ition of reaction path degeneracy, in terms of symmetry numbers {\i\f23 s} and the value of {\i n} is found in Gilbert and Smith.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b Note H }Sinusoidally-hindered rotors [Details in Jordan, Smith and Gilbert, {\i J}{\plain \i\f20 }{\i Phys. Chem.,} {\b 95}
, 8685 (1991)]. If JAV = 1, then the two-dimensional internal rotors that are used to model the rocking motions in loose transition states are treated as free rotations (modified if necessary by a hard-sphere hindrance). If JAV = 2, these rotations are tr
eated using a sinusoidal function {\i V}({\i\f23 q}) = {\fs16\up4 1}{\fs16 /}{\fs16\dn4 2}{\i V}{\fs14\dn4 0} (1 \endash cos{\i\f23 q}) for non-symmetric bonding fragments (e.g., OH), or:\par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab {\i V}({\i\f23 q}) = {\i V}{\fs14\dn4 0} (1 \endash cos{\i\f23 q}), 0 \'b2 {\i\f23 q}{\i }\'b2 \|F({\f23 p},2) ;\par
{\i \tab \tab V}({\i\f23 q}) = {\i V}{\fs14\dn4 0} (1 + cos{\i\f23 q}), \|F({\f23 p},2) \'b2 {\i\f23 q}{\i }\'b2 {\f23 p}\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 for symmetric fragments (e.g., planar CH{\fs18\dn4 3}). {\i V}{\fs14\dn4 0}
is the maximum amount of destabilization of the transition state caused by loss of orbital overlap as the moieties rotate, at a given {\i r}{\fs18\up6 \'a0}. {\i V}{\fs14\dn4 0} is approximated by {\plain \f20 |} {\i V}({\i r}{\fs18\up6 \'a0}) {\plain
\f20 |}, i.e., the maximum stabilization possible at {\i r}{\fs18\up6 \'a0}; here {\i V}({\i r}) is the interaction between the moieties (usually a Morse curve).\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 HIND is the parameter specifying the number of hindered internal rotors in the transition state. For a radical/radical reaction, this could be 1 or 2. For an ion/dipole reaction (e.g., CH{\fs18\dn4 3}{\fs18\up6 +}
+ HCN) HIND = 1; here the hindered rotor is the dipole rotating. If HIND is specified as negative, then hard-sphere steric interaction can be explicitly incorporated. The hindrance angles THETA1 and THETA2 give the angle through which the fragment can rot
ate, and are calculated using program GEOM. The symmetry of the fragments, ISYM1 and ISYM2
, must also be included. It is important in this case to enter the free (unhindered) rotor values for BVEC. The steric interaction should only be included for neutral/neutral reactions, since ion/dipole reactions have transition states located at very larg
e {\i r}{\fs18\up6 \'a0} where steric interactions (except the ion/dipole one) are insignificant.\par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
\pard \qj\ri29\sl280\tx720\tqc\tx3960 {\b Note I} By changing IXV and JXV in the program one can change the form of {\i P}({\i E,E'}\|D\|FO1()). The program has the following functional form for {\i P}({\i E,E'}\|D\|FO1()) (unless the biased ra
ndom walk option is used): \par
\pard\plain \s3\qj\ri29\sb140\sa140\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab {\i P}({\i E,E'}\|D\|FO1()) = \|F(1,{\i N}({\i E'})) {\i r}({\i E} \endash {\i E'}), {\i E}< {\i E'}\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 where {\i r}({\i E}) = {\i x}{\fs18\up6 IXV} exp(\|D\|FO1()\endash \|D\|FO1(){\i x}{\fs18\up6 JXV}), where {\i x} = |{\i E} \endash {\i E'}|/{\i\f23 a} and {\i E}<{\i E'} (the normalizing factor {\i N
}({\i E'}) is evaluated automatically by the program, and the values for {\i E} > {\i E'} are determined exactly from {\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 microscopic reversibility}}). The value(s) of the parameter {\i\f23 a}
are the ALPHAV. One can thus use this option to select a wide variety of functional forms for {\i P}({\i E,E'}\|D\|FO1()): e.g., a Gaussian if IXV=0, JXV=2; Poisson if IXV=1, JXV=
1. The (default) values of IXV and JXV in the output file prepared by RRKM are for an exponential form for {\i r}({\i E }\endash {\i E'}): i.e., IXV = 0, JXV = 1. In the case of an exponential form, <\'c6{\i E}{\fs18\dn4 down}> = {\i\f23 a}. \-
If one inputs both IXV and JXV as negative, then the program adopts the \ldblquote biased random walk\rdblquote model for {\i P}({\i E,E'}\|D\|FO1()) (see eq 12); the value of {\i\f23 a} is then taken to be the value of the parameter {\i s}
in that model.{\v {\xe\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 {\v biased random walk model}}}\par
\par
\pard \qj\ri29\sl-280\tx720\tqc\tx3960 {\b Note J} \-If the first pressure, PR(1), is input as a negative number, then the program generates an entire falloff curve at the NP pressures equally spaced in powers of 10 around the median pressure where [M]{
\i k}{\fs14\up6 0}/{\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}) = 1, where {\i k}{\fs14\up6 0}
is the low-pressure rate coefficient. This option is not available if the angular momentum conservation option is used. Note the following relationship between the true unimolecular rate coefficient {\i k}{\fs18\dn4 uni}, the high-pressure value {\i k}\|
s\|do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}) and the low-pressure limiting value [M]{\i k}{\fs14\up6 0}; one {\b always} has {\i k}{\fs18\dn4 uni} \'b2 [M]{\i k}{\fs14\up6 0} and {\i k}{\fs18\dn4 uni} \'b2 {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|s
\|up5({\fs14 \'b0}) .\par
\par
\pard \qc\ri29\sl280\tx720\tqc\tx3960 {{\pict\macpict\picw295\pich210\picscaled
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\pard \qj\ri29\sl280\tx720\tqc\tx3960 \par
{\b Note K} Concerning selection of (i) tolerances (ERR1, ERR2 in particular), (ii) of the final value of the grain size dE and (iii)\~of PALMT: a useful indication that one or more of these is too large is that either of the broadening factors {\i F}{
\fs14\dn4 WC} or {\i F}{\fs14\dn4 SC} exceeds unity, or that the collision efficiency {\i\f23 b}
exceeds unity (although if the amount by which any of these quantities exceed unity is only a few percent, no action is necessary). The program therefore prints out a warn
ing message if any of these exceed unity by more than 5%, as well as recommended action. Note also that a grain size of less than 100 cm{\fs18\up6 \endash 1}
is never required under normal circumstances, and indeed the program will select this as its final grain size irrespective of the input values. N.B. If the machine has a small word length and EXP rather than DEXP, etc., are used for functions, this warnin
g may also be given. It has been found that regions of numerical instability may arise when {\i E}{\fs14\dn4 0} is so low that the average thermal energy exceeds {\i E}{\fs14\dn4 0}, and <\'c6{\i E}{\fs18\dn4 down}
> is very small; the occurrence of this instability is indicated by any of the above messages and grossly different eigenvalues with decreasing grain size \'c9 no cure for this (rarely occurring) instability has yet been found.\par
\par
{\b Note L} The first two atoms must be assigned the following values: NB(1)=0; A(1)=0; B(1)=0; R(1)=0.; NB(2)=1; A(2)=0; B(2)=0; R(2)= appropriate bond length.\par
\par
{\b Note M} The program prints a warning message if any two atoms are closer than 0.9 \'81, whi
ch is the shortest likely chemical bond length. This is because a mistake in the dihedral angles is likely to manifest itself as two atoms lying almost on top of each other.\par
\par
\pard\plain \s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 {\b Note N} Use of program for canonical variation choice of transition state. For reactions where there is no barrier to {\v {\xe\pard\plain
\s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 recombination {\v (association) rate coefficient}}}(e.g., radical-radical recombinations or {\v {\xe\pard\plain \s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 ion-molecule{\v reaction}}}
associations), {\v {\xe\pard\plain \s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 {\v variational selection of transition state}}}the transition state is best chosen variationally, i.e., to give a minimum {\i k}\|s\|do2({\fs14 uni})\|d\|ba7()\|
s\|up5({\fs14 \'b0}) . To use the program package for canonical variation is simple. 1. Choose the transition state geometry for everything except {\i r}{\fs18\up6 \'a0}, the length of the breaking bond (the reaction coordinate). The value of {\i r}{
\fs18\up6 \'a0} will be chosen variationally. 2. Set up GEOM for the transition state, and run it with varying estimates of the length of the breaking bond in the transition state. 3. After running GEOM for a given {\i r}{\fs18\up6 \'a0}
, then run RRKM; to do this for each {\i r}{\fs18\up6 \'a0}, one changes only the following transition state parameters, each of which is printed out by GEOM: value of {\i B} for two-dimensional inactive external rotor; {\i B}
value for active external rotor; {\i B} values for the two 2-dimensional hindered rotors (these hindered {\i B} values are printed out directly by GEOM, and need no further calculation); {\i B}
value for torsion (also printed out by GEOM, the very last line). Note that RRKM should be run with active and inactive external rotors properly included [see Note C(v)]. 4. Choose {\i r}{\fs18\up6 \'a0} as that which value gives the lowest {\i k}\|s\|
do2({\fs14 uni})\|d\|ba7()\|s\|up5({\fs14 \'b0}) for a given temperature; note that {\i r}{\fs18\up6 \'a0} will vary with temperature.\par
\par
{\b Note O} This parameter truncates the maximum energy considered. For the case of {\v {\xe\pard\plain \s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 multichannel reactions}}
, it is necessary to ensure that ERR1 is sufficiently small that the upper channel is taken into account. For example, if there is a large difference between the critical energies if each channel, then if ERR1 is moderately large (e.g., the default value o
f 1E\endash 3) then the maximum energy considered nay be below the higher {\i E}{\fs14\dn4 0} : this will be indicated by a zero value of {\i k}{\fs18\dn4 uni} for the second channel. To cure this, simply make ERR1 much smaller (e.g., 1E\endash 10).{\b
\par
}\page {\v {\tc\tcl2\pard\plain \s4\qj\ri360\sl280\tx720\tqc\tx3960\tqr\tx7920 \f20\fs20 \tab \tab {\b\fs28 Index}}}\par
\pard\plain \qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 \par
\pard\plain \s240\qj\ri29\sl280\tx720\tqc\tx3960 \f20\fs20 angular momentum conservation 3, 4, 5, 6, 7, 11, 13, 18, 26, 27\par
biased random walk model 3, 5, 16, 29\par
collision frequency 8\par
dihedral angles 18\par
downward energy transferred per collision 4\par
electronic degeneracy 26\par
energy interconversion 25\par
energy transferred per collision 4\par
Fortran compiler 1\par
input description 6, 11, 14, 16\par
interconversion, energy 25\par
ion-molecule association 4\par
ion-molecule reaction 1, 5, 6, 7, 9, 30\par
J-conservation 3, 4, 5, 6, 7, 11, 13, 26\par
Lindemann-Hinshelwood mechanism 2\par
master equation 2, 6\par
mean energy transfer rate coefficient 4\par
mean energy transferred per collision 4\par
mean-square energy transferred per collision 4\par
microscopic reaction rate 3\par
microscopic reversibility 28\par
moment of inertia conversion factor 25\par
moments of inertia 3\par
multichannel reaction 11, 27\par
multichannel reactions 30\par
probability of energy transferred per collision 2\par
reaction path degeneracy 7, 9, 26\par
recombination (association) rate coefficient 1, 2, 4, 18, 23, 25, 26, 30\par
root mean-square energy transfer rate coefficient 4\par
rotations 25\par
RRKM theory 1, 3, 4, 5, 6, 14, 18\par
simple-fission transition state 4, 6, 7, 14, 18, 26\par
symmetry number 7, 9, 26\par
transition state theory 3\par
variational selection of transition state 4, 14, 18, 30\par
}
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