From maiden@RedBrick.DCU.IE Fri May 9 06:04:26 1997 Received: from tolka.dcu.ie for maiden@RedBrick.DCU.IE by www.ccl.net (8.8.3/950822.1) id EAA16396; Fri, 9 May 1997 04:57:37 -0400 (EDT) Received: from Nurse.RedBrick.DCU.IE by tolka.dcu.ie; (5.65v3.2/1.1.8.2/14Feb96-0535PM) id AA21145; Fri, 9 May 1997 09:55:40 +0100 Received: from localhost (maiden@localhost) by RedBrick.DCU.IE (SomeMTA) with SMTP id JAA01252 for ; Fri, 9 May 1997 09:57:09 +0100 (BST) Organisation: DCU Networking Society X-Disclaimer: The DCUNS is not responsible for the content of this message X-Award: Most Improved Society 1996-7 Date: Fri, 9 May 1997 09:57:09 +0100 (BST) From: Michael Nolan To: chemistry@www.ccl.net Subject: mn and cr complexes (Summary: part 2) Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII hi CCL!! here are the other excellent replies i recieved. answer 1: From kless@chem.ucla.edu Thu May 1 08:13:28 1997 Date: Wed, 30 Apr 1997 11:29:34 -0700 (PDT) From: Achim Kless To: michael nolan Subject: Re: CCL:mn complexes Dear Michael, > I have a really bad problem and I cannot do anything about it. > Here it is: > I am trying to run ab-initio calculations on Mn(CO)3 and Cr(CO)3, using > GAMESS-UK. > I have tried the following methods (I also give if ti was succesful or > not): > > 1) RHF/3-21G (3-21G on all atoms) > for Mn(CO)3 this is successful, for Cr(CO)3 I get excessive number of > iterations and an oscillation of the energy. > > 3) RHF with minimal basis ECP on all atoms for Mn(CO)3 and Cr(CO)3. > This gives a ludicrously low energy (about -70a.u.), oscillations in the > energy and excessive number of iterations > > 5) RHF, min. basis ECP and ECPDZ on Mn, 3-21G and 6-31G on C and O try out the basis sets from A. Schaefer, C. Huber, and R. Ahlrichs. They are suitable especially for the middle-transition metals. See: Fully Optimized Contracted Gaussian Basis Sets of Triple Zeta Valence Quality for Atoms Li to Kr. A. Schaefer, C. Huber, and R. Ahlrichs; J. Chem. Phys. 100, 5829 (1994). Regarding ECPs check out the Pseudopotentials from M. Dolg, H. Stoll, H. Preuss. (http://www.theochem.uni-stuttgart.de) best regards, Achim --------------------------- Dr. Achim Kless UCLA, Dept. Chemistry and Biochemistry This was very helpful, I found these bais sets useful and also Dunning's double-zeta basis sets. ******************************************************************************* answer 2: From her10531@argon.chem.tu-berlin.de Thu May 1 08:13:36 1997 Date: Wed, 30 Apr 1997 19:51:16 +0200 From: Rolad Hertwig To: michael nolan Subject: Re: CCL:mn complexes Micheal, welcome to the world of transition metal quantum chemistry! I will try to give you some hints, but in addition you should get in touch with people ( I mean personally), who have experience in the field, if your thesis advisor has not. > 1) RHF/3-21G (3-21G on all atoms) > for Mn(CO)3 this is successful, for Cr(CO)3 I get excessive number of > iterations and an oscillation of the energy. Oscillation is not unusual in organo-metallic complexes. Play around with SCF accelerators, like damping, diis, and shifting of virtual orbitals. There is no general recipe to avoid this. Also, change your starting geometry. Avoid too small bond lengths. Exploit symmetry, and specify electronic occupations explicitly as far as this is possible (I have little experience with GAMESS). > 3) RHF with minimal basis ECP on all atoms for Mn(CO)3 and Cr(CO)3. > This gives a ludicrously low energy (about -70a.u.), oscillations in the > energy and excessive number of iterations ECPs emulate the shielding of core electrons. Since these electrons are not treated explicitly, they do not contribute to the total energy, therefore you get small total energies. Do not compare total energies calculated with ECPs with those resuting from all electron calcs. > Does anyone have any ideas for how I can use ECPs (we are limited to about > 1.2GB) or anything else to get these calculations to run (we do not yet > have DFT). Okay, ECPs are not t h e solution to your problems as long as you do not have to worry about relativistic effects (Not crucial for 3d TMs). If you use them, be sure to use the basis set that comes along with them, since it has been specifically designed for that certain ECP. You do not necessarily need more than 1 GB of disk space, unless you are going to use correlated post-HF methods. DFT would be suitable for your case, since HF is rather inappropriate. However it is not te worst to start with. I hope I could give you some general clues, but it is not easy to do this via email for the problems you have. I would really recommend, you rip some money out of yor advisor's pockets and travel somewhere for a couple of weaks to learn the trade. Good luck! Roland ----------------------------------------------------------------------- Roland H. Hertwig Institut fuer Organische Chemie, TU-Berlin ******************************************************************************** answer 3: From Philippe.Maitre@cth.u-psud.fr Thu May 1 08:13:45 1997 Date: Wed, 30 Apr 1997 18:55:26 +0200 From: Philippe Maitre To: Maiden@RedBrick.DCU.IE Subject: your Mn and Cr complexes, Dear Michael, I have had similar problems and I guess I could help you. You said "I assume the fact that Mn(CO)3 is positively charged has nothing to do with it (it is a closed shell)". I do not agree. The typical problem of SCF convergence with Transition Metal is that you have a near generacy of several d orbitals for low-coordination complexes. This leads to a near degeneracy of electronic configurations, which therby leads to a near degeneracy of determinants. And that's your problem in your iteration. This problem can be further complicated, sometimes, by pseudo potential. I do not have any experience with Gamess-UK, but with a Gaussian92 or Gaussian94, the internal guess of orbitals (either generated by diagonalizing HCORE or by using the extended Huckel approach) is extremely bad when using Pseudo-potential. Do not ask me why, I do not know. Just one more thing : this bad initial guess obtained when using a pseudo potential does not mean at all that the pseudo potentials are bad. There is a simple trick to converge an scf calculation with low-coordinationtransition metal system. You take a piece of paper and you find out the orbital energy level. For ML3 systems, you can look at "Orbital interaction in chemistry" by Albright, Burdett and Whangbo. Then, you find out a spin state (with the same numebr of electrons or not) which leads to a large energy difference between the Ground state electronic configuration and the first electronic one (i.e. where there is a large gap between the HOMO and the LUMO). Run an ROHF or RHF if this is a closed shell on this guy. This will converge easily. One it's done, reuse the converged orbitals of this calculation to calculate your system of interest. Good Luck, If you have any problem, do not hesitate to contact me. But please, give me a geometry and a spin state. Philippe Maitre Laboratoire de chimie theorique Batiment 490 Universite de Paris XI 91405 Orsay , FRANCE ************************************************************************* answer 4: From tcundari@msuvx2.memphis.edu Thu May 1 08:13:52 1997 Date: Wed, 30 Apr 1997 11:57:30 -0600 From: Tom Cundari To: michael nolan Subject: Re: CCL:mn complexes Dear Michael, Noticed you said something about an RHF calc. What is the multiplicity of Cr(CO)3 and Mn(CO)3 fragments you are trying to converge on? Are you using symmetry? Another trick you can try with ECPs is to do a quick first calc on the 2nd row analogue (Mo for Cr, Tc for Mn) and if this converges use this wavefunction to start off a subsequent calculation of the first row metal. 2nd row transition metals often behave better in the SCF since their bonding is more covalent and hence the ligand fields are typically larger. At any rate, with ECPs the number of electrons should be the same and the symmetry props of the MOs should be similar. Good luck. Tom Tom Cundari Associate Professor Department of Chemistry University of Memphis (under T in the ACS Grad Directory!) ***************************************************************************** answer 5: From: chemistry-request (SMTPMAIL.chemistr) at PROFGATE Date: 4/30/97 9:57AM To: Michelle Pietsch at BVL60PO *To: C=us; A= ; P=Internet; DD.RFC-822=chemistry(a)www.ccl.net at X.400 Subject: CCL:mn complexes ------------------------------------------------------------------------------- Michael, To force convergence, increase the value of LEVEL (keyword for GAMESS-UK). You may increase this value until you force convergence. Once the wave function is converging, decrease the value for LEVEL little by little until it is at its default value. Using a large value for LEVEL will force convergence but it may converge in the wrong state. After convergence, print out your molecular orbitals and make sure that the correct orbitals are occupied. If the correct orbitals are not occupied, use the SWAP command to exchange occupied and virtuals until the correct orbitals are occupied. Always check to make sure that you have the correct orbitals occupied after using the SWAP command. You may want to examine the molecular orbitals before convergence and make any SWAPs at that time. I have found this procedure to work well for systems containing Cr and other metals. You should be able to use ECPs and any basis set. However, increasing the number of valence functions will increase the difficulty of convergence. Best of Luck, Mickey ******************************************************************************* answer 6: From bianco@lord.Colorado.EDU Thu May 1 08:14:04 1997 Date: Wed, 30 Apr 1997 09:13:44 -0600 (MDT) From: Roberto Bianco To: michael nolan Subject: Re: CCL:mn complexes Hi Michael, looking at the Metal-C distance of 1.8 angstrom you give below, my guess is that the atomic centers are too far apart for the wavefunction to converge. I would try with initial, artificially shorter Metal-C distances (say 1.2-1.4 A) to achieve convergence, and then let the geometry relax (optimize) through small steps, such that the "memory" of the converged wavefunction is not lost in the process. Roberto -- Roberto Bianco / Department of Chemistry & Biochemistry University of Colorado / Campus Box 215 / Boulder, CO 80309 / USA bianco@lord.colorado.edu / phone +(303) 492-3504 / fax +(303) 492-5894 ************************************************************************** answer 7: From E.A.Moore@open.ac.uk Fri May 9 09:55:50 1997 Date: 1 May 1997 10:08:10 +0100 From: "E.A.Moore (Elaine Moore)" To: michael nolan Subject: Mn and Co complexes Transition metal complexes are vey tricky. You can try 1) Altering the guess option 2) Putting damping on to converge and then feeding the resulting orbitals back in with the damping off 3) reordering the orbitals ECP energies will be smaller because of the missing core electrons. Elaine A. Moore e.a.moore@open.ac.uk ***************************************************************************** Michael Der Kobold hat gesprochen ****************************************************************************** Michael Nolan (nearly BSc.) Dublin City University (quantum chem.) 41 Woodview Chemistry + German Year 4 Lucan Co. Dublin Ireland / Republic of Ireland / ROI / Eire Email: Maiden@redbrick.dcu.ie ****************************************************************************** From fiona@rsc.anu.edu.au Fri May 9 06:41:59 1997 Received: from anugpo.anu.edu.au for fiona@rsc.anu.edu.au by www.ccl.net (8.8.3/950822.1) id GAA17038; Fri, 9 May 1997 06:10:50 -0400 (EDT) Received: from rsc (rsc.anu.edu.au [150.203.35.129]) by anugpo.anu.edu.au (8.8.5/8.8.5) with SMTP id UAA18157 for ; Fri, 9 May 1997 20:10:36 +1000 (EST) Received: from [150.203.37.11] by rsc (SMI-8.6/SMI-SVR4) id UAA16833; Fri, 9 May 1997 20:16:39 +1000 Date: Fri, 9 May 1997 20:16:39 +1000 Message-Id: <199705091016.UAA16833@rsc> Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: chemistry@www.ccl.net From: fiona@rsc.anu.edu.au (Dr Fiona Bettens) Subject: Anisotropic Hyperfine Summary Dear Netters, My original query was: >Is anyone aware of any software that will either take the >molecular-orbital expansion coefficients produced in an ab initio >calculation or do the ab initio calculation itself to calculate the >expectation values of the anisotropic hyperfine tensor. By this I >mean the tensor that represents the coupling of the electronic >mangnetic dipole with the nuclear magnetic dipole. I am interested in >transition metal complexes with more than one unpaired electron, this >complicates matters somewhat. I suspect MELDF does what I want, but >it is difficult to find documentation on exactly how it does it. ******************* ******************* The anisotropic hfs is proportional to the efg at the nucleus produced by a hypothetical charge density representing the spin density. Therefore in G94 the input card #P cisd/6-31g** scf=direct density=all IOP(6/26=4,6/17=2) prop=efg is sufficient to generate the anisotropic hf tensor for all nuclei, subject to the appropiate proportionality factors (available from a standard textbook). Sincerely Rod Macrae ******************* ******************* First note that in my reply I am \underline{not} talking about the nuclear quadrupole splitting, which is evaluated by calculating the true efg at the nucleus (i.e. not a spin property) and multiplying it by a nuclear term extracted from a spin hamiltonian treatment of the nuclear component of the interaction. However, this does give a guide as to how to think about the nucleus-electron dipole-dipole hyperfine interaction. The expression will contain an operator (describing the angular and radial properties of the interaction) evaluated at the nucleus. This will be multiplied by a factor in which the nuclear part of the interaction, expressed in terms of a spin hamiltonian, will be embedded. The efg operator takes the form $(3 \cos^{2}\theta_{J}-1)/r^{3}$. (I will write mathematical expressions in Latex form so that you can print them out and look at them if you want.) The nuclear quadrupole coupling has electronic component $V_{zz} = {\langle} \psi_{e} | {\cal V}_{zz} | \psi_{e} {\rangle}$. In the case of the dipolar hyperfine interaction the angular expression in terms of the nuclear-electronic spin hamiltonian is given by (Atkins 14.11.2): \begin{displaymath} \hat{\cal H}_{hf} = \mu_{0} g_{e} \gamma_{e} \gamma_{N} \frac{({\bf s}\cdot{\bf I} - 3 {\bf s}\cdot \hat{\bf r} \hat{\bf r} \cdot {\bf I})}{r^{3}}. \end{displaymath} At this point what generally happens is that a \underline{high field approximation} is made (aligned spins), and all spin components other than the z components disappear. What this explicitly means is that the Zeeman interaction is much larger than the hyperfine interaction, and only the largest terms in the expansion of the dipole-dipole interaction - see e.g. Abragam p. 104 - need be considered (as a perturbation of the Zeeman interaction). It is this procedure that leads to a dipole-dipole coupling proportional to $\frac{1}{r^{3}} (1 - 3 \cos^{2}\theta ) I_Z s_Z$, and therefore essentially proportional to an "electric field gradient" constructed from the spin density (rather than the charge density) distribution. This is the conceptual procedure behind my previous answer. I don't know offhand if 1) this approximation procedure is less valid for I > 1/2, or 2) there is a commercially-available program which performs the calculation using the full expression. (If you find one, I'd very much appreciate it if you'd tell me.) Sincerely Rod Sources: Atkins, Quantum Mechanics, Lucken, Nuclear Quadrupolar Interactions, Abragam, Principles of Nuclear Magnetism. R. M. Macrae, Muon Science Laboratory Institute of Physical and Chemical Research (RIKEN) e-mail: macrae@rikaxp.riken.go.jp (normal) and : macrae@rikmtl.riken.go.jp (MIME-encoded) Tel : (81) 484 62 1111 ext 3336 Fax : (81) 484 62 4648 ******************* ******************* The properties package in MELDF deals with some basic expectaction values, e.g. del(A)s, the del operator on center A evaluated over the spin density. For the anisotropic components of the hyperfine interaction it needs operators like xx/r**5, xy/r**5, yy/r**5, etc. also evaluated over the spin operator. The latter values are combined to form the anisotropic hyperfine (3x3) matrix in the molecular axis system, using a convention that requires the trace of this matrix to have a zero trace. Some of the components are defined as follows: A(xx) = (3/2)*(x2-y2)/r5 - (0.5)*(z2-r2)/r5 A(yy) = -(3/2)*(x2-yy)/r5 - (0.5)*(z2-r2)/r5 A(zz) = (z2-r2)/r5 Ernest Davidson and I have written a chapter in a book on the general topic of computing hyperfine parameters. It's in Theoretical Models of Chemical Bonding, Part 3, Spinger Verlag, 1991. David Feller | Environmental Molecular Sciences Laboratory | Battelle Pacific Northwest National Lab | Mail Stop K1-90 | e-mail:d3e102@emsl.pnl.gov 906 Battelle Blvd | Fax: (509)-375-6631 Richland, WA 99352 | ******************* ******************* The folks at Gaussian should have responded by now, but they haven't, and I've given up on them. But I think that G94 really does calculate the EFG tensor properly. IOp(6/17=2) selects the spin density, rather than the charge density, and IOp(6/26=4) causes only the electronic component of the integrals to be calculated, which is what we want. I have used the EFG to calculate the anisotropic coupling for the methyl radical as was done by Adamo, Barone, and Fortunelli in J. Chem. Phys. 102 (1995) 384, and I can reproduce their values almost exactly ( I think they used a slightly different B3LYP functional ). I then calculated the couplings for an organometallic ion that I've been working on, and I get values that are within 4-18% of experiment (which, incidentally, is the same margin of error as reported by Adamo, et al. for the methyl radical). This looks pretty good, but I'm trying to improve the results. The formula I used is: g B g_n B_n dE T_ii = ----------- -- a^3 h 10^13 dq_i where g and g_n are the electron and nuclear g factors, B and B_n are the corresponding magnetons, a is the bohr radius, and since the tensor is already diagonalized, only the _ii elements are there. G94 reports dE/dq_i in atomic units, which means (1/bohr^3), and I've converted this to MHz for T_ii with the terms in the denominator (there is a factor of 10^7 in the denominator, too, from the permeability constant). Dale Braden Department of Chemistry University of Oregon Eugene, OR 97403-1253 genghis@darkwing.uoregon.edu ******************* ******************* Here is Dan Chipman's response to my query about the problem with the x and y components of the spin operators, spin contamination, etc. Dale Braden ---------- Forwarded message ---------- Dale, It does indeed appear that G94 will do what you want by setting the options you specify. Even so, I would suggest that you make a check by trying to reproduce some number in the literature. I have found that the G94 manual is not always in concordance what what actually happens in the code. The x,y,z indices on the S_x, S_y, and S_z spin operators refer to the components relative to the applied magnetic field, which are generally unrelated to the X,Y,Z coordinates of atoms in the molecule-fixed frame. Usually, S_x and S_y only produce small second-order effects in the hyperfine interaction. For the normal first-order effects, one only need consider S_z. For a good discussion of this, look at the old but good textbook "Introduction to Magnetic Resonance" by Carrington and MacLachlan. "Averaging over spin density" is a somewhat different concept than what you indicate. It means taking the difference between values obtained from the sum of all alpha-spin orbitals and the sum of all beta-spin orbitals. Averaging over the total density, on the other hand, means adding the alpha and beta contributions instead of subtracting them. Unfortunately, taking account of spin contamination is a difficult problem. There is no simple "renormalization" correction that would take care of it. Some people suggest projecting the incorrect spin terms out of the wave function, but that is not really satisfactory either. For isotropic hyperfine interactions, neither unrestricted-Hartree Fock nor any of its spin-projected variants is reliable. However, anisotropic hyperfine interactions are often dominated by the singly-occupied molecular orbital in which case UHF, PUHF, etc. may all work fairly well. That is, when the contribution of the SOMO is dominant, one need not worry about spin contamination effects. If spin contamination is a problem, then one can either resort to spin-restricted approaches, as I usually do, or to correlated methods based on UHF such as UQCISD adn UQCISD(T) that automatically remove most of the spin contamination. Unfortunately, these latter methods are quite expensive. Dan ******************* ******************* I have received several requests about the computation of ESR anisotropic coupling constants by Gaussian94. Since this seems a general question I directly post this message for the whole CCL community. These constants are nothing else than field gradients computed with the spin rather than the total electronic density and not including the nuclear contribution. Furthermore the resulting tensor must be put in the zero-trace form. In gaussian94 it is possible to force these options by setting in the keyword list the following items: PROP IOP(6/17=2,6/26=4) Here follows an input and the relevant part of the output for a STO-3G computation of H2NO ---------------------------------------------------------------------- #UHF/PROP IOP(6/17=2) IOP(6/26=4) H2NO 0 2 X X 1 1.0 N 2 1.0 1 90.0 H 3 NH 2 90.0 1 THETA H 3 NH 2 90.0 1 -THETA O 3 NO 2 ALPHA 1 180.0 NH=1.0179 NO=1.2778 THETA=58.9235 ALPHA=110.0 ---------------------------------------------------------------------- Fermi contact analysis (atomic units). 1 1 N .042923 2 H -.005038 3 H -.005038 4 O .098025 ********************************************************************** Electrostatic Properties Using The SCF Density ********************************************************************** Warning! Using spin rather than total density! --- Only the electronic contributions will be computed --- ----------------------------------------------------- Center ---- Electric Field Gradient ---- XX YY ZZ ----------------------------------------------------- 1 Atom .119043 -.295145 -.363285 2 Atom .002506 .031422 .029384 3 Atom .002506 .031422 .029384 4 Atom 2.130337 -1.663300 -1.698867 ----------------------------------------------------- ----------------------------------------------------- Center ---- Electric Field Gradient ---- ( tensor representation ) 3XX-RR 3YY-RR 3ZZ-RR ----------------------------------------------------- 1 Atom .298838 -.115349 -.183489 2 Atom -.018598 .010318 .008280 3 Atom -.018598 .010318 .008280 4 Atom 2.540947 -1.252690 -1.288257 ----------------------------------------------------- these are the principal values of anisotropic coupling constants ---------------------------------------------------------------------- Since the directions of principal moments are often significant and transforma- tion to more conventional units can be performed once for ever, I have modified the links 601 and 602 of gaussian to obtain the following output for the same input ---------------------------------------------------------------------- ---------------------------------------------------------------------- Isotropic Fermi Contact Couplings ---------------------------------------------------------------------- Atom a.u. MegaHertz Gauss 10(-4) cm-1 1 N(14) .04292 13.86856 4.94865 4.62605 2 H -.00504 -22.51979 -8.03562 -7.51179 3 H -.00504 -22.51979 -8.03562 -7.51179 4 O(17) .09802 -59.42481 -21.20426 -19.82198 ---------------------------------------------------------------------- ********************************************************************** Electrostatic Properties Using The SCF Density ********************************************************************** Warning! Using spin rather than total density! --- Only the electronic contributions will be computed --- Atomic Center 1 is at -.021142 .543231 .000000 Atomic Center 2 is at .158563 1.036966 .871810 Atomic Center 3 is at .158563 1.036966 -.871810 Atomic Center 4 is at -.021142 -.734569 .000000 ----------------------------------------------------- Center ---- Spin Dipole Couplings ---- 3XX-RR 3YY-RR 3ZZ-RR ----------------------------------------------------- 1 Atom .298838 -.115349 -.183489 2 Atom -.018598 .010318 .008280 3 Atom -.018598 .010318 .008280 4 Atom 2.540947 -1.252690 -1.288257 ----------------------------------------------------- Center ---- Spin Dipole Couplings ---- XY XZ YZ ----------------------------------------------------- 1 Atom -.074772 .000000 .000000 2 Atom .004800 .007098 .035617 3 Atom .004800 -.007098 -.035617 4 Atom -.099769 .000000 .000000 ---------------------------------------------------------------------- Anisotropic Spin Dipole Couplings in Principal Axis System ---------------------------------------------------------------------- Atom a.u. MegaHertz Gauss 10(-4) cm-1 Axes Baa -.1835 -7.077 -2.525 -2.361 .0000 .0000 1.0000 1 N(14) Bbb -.1284 -4.953 -1.767 -1.652 .1724 .9850 .0000 Bcc .3119 12.030 4.293 4.013 .9850 -.1724 .0000 1/R**3 -.5394 -20.803 -7.423 -6.939 Baa -.0268 -14.276 -5.094 -4.762 -.2361 -.6571 .7158 2 H Bbb -.0193 -10.278 -3.667 -3.428 .9631 -.2560 .0828 Bcc .0460 24.554 8.762 8.190 .1288 .7090 .6933 1/R**3 .0633 33.780 12.053 11.268 Baa -.0268 -14.276 -5.094 -4.762 .2361 .6571 .7158 3 H Bbb -.0193 -10.278 -3.667 -3.428 .9631 -.2560 -.0828 Bcc .0460 24.554 8.762 8.190 .1288 .7090 -.6933 1/R**3 .0633 33.780 12.053 11.268 Baa -1.2883 93.221 33.264 31.095 .0000 .0000 1.0000 4 O(17) Bbb -1.2553 90.837 32.413 30.300 .0263 .9997 .0000 Bcc 2.5436 -184.059 -65.677 -61.395 .9997 -.0263 .0000 1/R**3 -1.2318 89.138 31.807 29.733 ---------------------------------------------------------------------- ---------------------------------------------------------------------- Vincenzo Barone | Professor of Theoretical Chemistry | Dipartimento di Chimica | tel. +39-81-5476503 Universita' Federico II | fax +39-81-5527771 via Mezzocannone 4 | e-mail ENZO@CHEMNA.DICHI.UNINA.IT I-80134 Napoli | Italy | ______________________________________________________________________ Below is a LaTeX summary of my findings to the original question: **************************************************** \documentstyle{article} \begin{document} \title{Calculation of the Anisotropic Dipolar Coupling Constants} \author{Fiona L. Bettens and Ryan P. A. Bettens} \date{26th March 1997} \maketitle \section{Macroscopic Hamiltonian} The anisotropic hyperfine coupling constants, $A_{\tau\upsilon}$, are phenomenological constants utilized in a ``so called'' spin Hamiltonian, below (see [\ref{bib:car67}] for example). This Hamiltonian is written for one nucleus of nonzero spin, $I_{\upsilon}$, \begin{equation} \hat{\cal H}_{\rm macro} = \mbox{\bf I}^{\dagger} \mbox{\bf A} \mbox{\bf S} = \sum_{\tau\upsilon} A_{\tau\upsilon} \hat{I}_{\tau} \hat{S}_{\upsilon} \end{equation} \noindent where the $\tau$ and $\upsilon$ represent the spaced fixed $X$, $Y$ and $Z$ axes and the operators $\hat{S}_{\upsilon}$ and $\hat{I}_{\tau}$ represent the components of {\it total} electronic spin and nuclear spin respectively. The spin Hamiltonian matrix is set up using the product basis, \begin{equation} |S, M_{S}; I, M_{I}\rangle = |S, M_{S}\rangle |I, M_{I}\rangle . \label{macro2} \end{equation} \section{Microscopic Hamiltonian} The Hamiltonian describing the {\it classical} interaction energy between a nuclear magnetic dipole and the magnetic dipoles of $n$ electrons is [\ref{bib:bev71}], \begin{equation} \hat{\cal H}_{\rm micro} = -\frac{\mu_{0}}{4\pi}g\beta g_{N}\beta_{N} \sum_{i=1}^{n} \left\{ \frac{{\bf I}\cdot{\bf s_{i}}}{{r_{i}}^{3}} - \frac{3({\bf I}\cdot{\bf r_{i}})({\bf s_{i}}\cdot{\bf r_{i}})} {{r_{i}}^{5}} \right\} \label{micro1} \end{equation} \noindent where the dimensionless constants $g$ and $g_{N}$ are the electronic and nuclear $g$ factors respectively; $\beta$ and $\beta_{N}$ are electronic Bohr and nuclear magnetons. The vectors {\bf I}, ${\bf s_{i}}$ and ${\bf r_{i}}$ represent the nuclear spin, electronic spin for electron $i$, and electronic position for electron $i$ with respect to the nucleus, operators. When Eq.\ (\ref{micro1}) is expanded the microscopic Hamiltonian becomes, \begin{equation} \hat{\cal H}_{\rm micro} = -\frac{\mu_{0}}{4\pi}g\beta g_{N}\beta_{N} \sum_{i=1}^{n} \left( \frac{1}{{r_{i}}^{3}}\right) \sum_{\tau,\upsilon}I_{\tau}s_{i\upsilon} \left( \frac{{r_{i}}^{2}\delta_{\tau\upsilon} - 3\tau_{i}\upsilon_{i}} {{r_{i}}^{2}} \right) . \label{micro2} \end{equation} If we consider here a single determinant electronic wavefunction, $\Psi_{q}$, {\it i.e.}, one for a given electronic configuration and labeled here with a $q$, then we can form a product basis with the nuclear spin eigenfunctions which corresponded to the eigenkets given in Eq.\ (\ref{macro2}). The resulting basis vectors are given by, \begin{equation} |q, M_{S}; I, M_{I}\rangle = |q, M_{S}\rangle |I, M_{I}\rangle . \label{micro3} \end{equation} \noindent The basis functions corresponding to these kets are operated upon by the microscopic Hamiltonian, Eq.\ (\ref{micro2}). \section{Equating the Expectation Values} In order to determine the anisotropic hyperfine coupling constants, $A_{\tau\upsilon}$, we compare identical matrix elements of the macroscopic and microscopic Hamiltonian matrices and equate like terms. Here we concern ourselves with only those terms diagonal in $S$ and $I$. Thus we have, \begin{eqnarray} & & \sum_{\tau\upsilon}A_{\tau\upsilon}\langle S, {M_{S}}'; I, {M_{I}}'| I_{\tau}S_{\upsilon} |S, {M_{S}}; I, {M_{I}}\rangle \nonumber \\ & = & -\frac{\mu_{0}}{4\pi}g\beta g_{N}\beta_{N} \sum_{i=1}^{n} \langle q, {M_{S}}'; I, {M_{I}}'| \left( \frac{1}{{r_{i}}^{3}}\right) \nonumber \\ & & \times \sum_{\tau,\upsilon}I_{\tau}s_{i\upsilon} \left( \frac{{r_{i}}^{2}\delta_{\tau\upsilon} - 3\tau_{i}\upsilon_{i}} {{r_{i}}^{2}} \right) |q, M_{S}; I, M_{I}\rangle \label{equate1} \end{eqnarray} \noindent so equating terms we obtain, \begin{equation} \langle S, {M_{S}}'|S_{\upsilon}|S, M_{S}\rangle A_{\tau\upsilon} = k \langle q, {M_{S}}'| \sum_{i=1}^{n} \left( \frac{1}{{r_{i}}^{3}}\right) \sum_{\tau,\upsilon}s_{i\upsilon} \left( \frac{{r_{i}}^{2}\delta_{\tau\upsilon} - 3\tau_{i}\upsilon_{i}} {{r_{i}}^{2}} \right) |q, M_{S}\rangle \label{equate2} \end{equation} \noindent where, \begin{equation} k = -\frac{\mu_{0}}{4\pi}g\beta g_{N}\beta_{N} \label{equate3} \end{equation} \noindent and we have facilitated this comparison by choosing eigenkets for nuclear spin states which have nonzero $M_{I}$, and, of course, a nonzero $I$. The nuclear dipole-electronic dipole interaction is described by the one electron operator given in Eq.\ (\ref{micro1}). When evaluating the expectation value of such an operator when the determinantal wavefunctions representing the bras and kets do not differ, {\it i.e.}, when $q$ and $M_{S}$ are the same in each bra and ket, the following expression can be shown to hold [\ref{bib:sza82}]. \begin{eqnarray} |q\rangle & = & |\cdots lm\cdots\rangle \label{oneelec1} \\ \langle q|\hat{{\cal O}}_{1}|q\rangle & = & \sum_{l}^{n}\langle l|h|l\rangle \label{oneelec2} \end{eqnarray} \noindent where $\hat{{\cal O}}_{1}$ is the one electron operator and is given by, \begin{equation} \hat{{\cal O}}_{1} = \sum_{i=1}^{n}h(i) . \label{oneelec3} \end{equation} \noindent and the $l$ and $m$ labels denote the spinorbitals $\chi_{l}$ and $\chi_{m}$. From Eq.\ (\ref{oneelec1}) - (\ref{oneelec3}), we obtain for Eq.\ (\ref{equate2}) [\ref{bib:bev71}], \begin{equation} A_{\tau \upsilon} = \frac{k}{2\langle S_{Z}\rangle} \sum_{\mu,\nu}^{N}\rho_{\mu\nu} \langle \mu | {r_{i}}^{- 5}({r_{i}}^{2}\delta_{\tau \upsilon} - 3\tau_{i} \upsilon_{i}) |\nu\rangle \label{equate4} \end{equation} \noindent where $\mu$ and $\nu$ are basis functions and the double sum is taken over all of them. The $\rho_{\mu\nu}$ is known as the spin density matrix and is given by the difference in the alpha and beta density matrices. \begin{eqnarray} \rho_{\mu\nu} & = & {P_{\mu\nu}}^{\alpha} - {P_{\mu\nu}}^{\beta} \\ {P_{\mu\nu}}^{\alpha} & = & \sum_{i=1}^{p}{c_{\mu i}}^{\alpha} {c_{\nu i}}^{\alpha} \\ {P_{\mu\nu}}^{\beta} & = & \sum_{i=1}^{q}{c_{\mu i}}^{\beta} {c_{\nu i}}^{\beta} \end{eqnarray} \noindent where the alpha and beta sums are over the $p$ occupied alpha orbitals and the $q$ occupied beta orbitals respectively. The $c_{\mu i}$ are the molecular orbital expansion coefficients for basis functions $\mu$ and MO $i$. Eq.\ (\ref{equate4}) is known as averaging the given orbital operator over spin density. It should be noted that the spin density matrix is intrinsicly different to the charge density matrix, which is the sum of the alpha and beta density matrices. It is of interest that elements of the electric field gradient are calculated in the same way as above except the average is taken over charge density. \section*{Bibliography} \begin{enumerate} \item \label{bib:car67} A. Carrington and A. D. McLachlan, {\it Introduction to Magnetic Resonance} (Harper and Row, New York, 1967). \item \label{bib:bev71} D. L. Beveridge and J. W. McIver, Jr., J. Chem. Phys. {\bf 54}, 4681, (1971). \item \label{bib:sza82} A. Szabo and N. S. Ostlund, {\it Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory} (MacMillian, New York, 1983). \end{enumerate} \end{document} Fiona Bettens The Australian National University, Email: Fiona@rsc.anu.edu.au Department of Chemistry, Fax: 61 6 249 0760 Canberra, ACT, 0200, AUSTRALIA From ccl@www.ccl.net Fri May 9 12:53:10 1997 Received: from bedrock.ccl.net for ccl@www.ccl.net by www.ccl.net (8.8.3/950822.1) id MAA23263; Fri, 9 May 1997 12:20:28 -0400 (EDT) Received: from cliff.acs.oakland.edu for bbesler@ouchem.chem.oakland.edu by bedrock.ccl.net (8.8.3/950822.1) id MAA25930; Fri, 9 May 1997 12:20:28 -0400 (EDT) Received: from ouchem.chem.oakland.edu (ouchem.chem.oakland.edu [141.210.108.5]) by cliff.acs.oakland.edu (8.8.5/8.7.3) with SMTP id MAA15279 for ; Fri, 9 May 1997 12:21:15 -0400 (EDT) Received: by ouchem.chem.oakland.edu; id AA14132; Fri, 9 May 1997 12:21:03 -0400 From: "Brent H. Besler" Message-Id: <9705091621.AA14132@ouchem.chem.oakland.edu> Subject: Non-DEC Alpha systems To: chemistry@ccl.net Date: Fri, 9 May 1997 12:21:03 -0500 (EDT) X-Mailer: ELM [version 2.4 PL21] Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Our University has a site license for DEC Unix and DEC compilers which covers non-DEC supplied Alpha systems. It is possible to purchase an Alpha system with 512 MB of RAM, 4GB of hard disk space, and 2 MB of static RAM cache from Microway for under $15,000. Has anyone had positive or negative experiences with the Microway systems? Also, I notice that DEC allows up to 8 MB of static RAM cache on their 500 Mhz Alphas. The motherboard diagram on www.microway.com indicated the Microway system could take 8 MB of cache. Does anyone have experience with the speed difference in the 500 Mhz Alpha with various cache sizes?