From aiba@ir.phys.chem.ethz.ch Fri Mar 31 11:39:58 1995 Received: from bernina.ethz.ch for aiba@ir.phys.chem.ethz.ch by www.ccl.net (8.6.10/930601.1506) id LAA08256; Fri, 31 Mar 1995 11:39:52 -0500 Received: from volta (actually volta.ethz.ch) by bernina.ethz.ch with SMTP inbound; Fri, 31 Mar 1995 18:39:29 +0200 Date: Fri, 31 Mar 1995 18:38:56 +0200 Message-Id: <95033118385669@ir.phys.chem.ethz.ch> From: aiba@ir.phys.chem.ethz.ch (Ayaz Bakasov, Phys. Chem., ETH Zurich) To: CHEMISTRY@ccl.net Subject: complex wave functions X-VMS-To: smtp%"CHEMISTRY@ccl.net" X-VMS-Cc: AIBA On March 30, 1995, Dr. Herbert H. H. Homeier wrote in response to my posting about complex wave functions: > One comment here: The basic reason > for using real wavefunctions/basis sets > is probably the theorem that in a system > with time reversal symmetry one can choose > the wavefunction to be real > IF spin is not considered. > A similar statement including spin > probably holds if closed shell systems are > considered (if somebody has a proof or > knows a reference, please send me an e-mail). I feel I can elaborate this point and put some textbook references on. Let me first cite L.D.Landau and E.M.Lifschitz [L&L]: "Quantum mechanics, Non-relativistic Theory", (Pergamon Press, Oxford, 1977, 3rd edition). [L&L]Page 55: ^^^^^^^^^ "Schroedinger equation for the wave functions \psi of stationary states are real, as are the conditions imposed on its solution. Hence its solutions can always be taken real. Footnote \dagger: These assertions are NOT valid for systems in magnetic field." "..the equations of quantum mechanics are unchanged by time reversal.. the invariance of the wave function when the sign of t is reversed and \Psi is simultaneously replaced by \Psi^* Footnote \ddagger: It is assumed that the potential energy U does NOT depend explicitly on the time: the system is either closed or in constant (NON-MAGNETIC) field". [L&L]Page 221: "..when spin is present, a refinement is necessary." [L&L]Page 222: "..not only in a closed system but in any external electric feild (theere being NO magnetic field) there is the symmetry with respect to the time reversal." What we see from this 'simple' statements is this: Schroedinger equation for stationary states does not depend on time, and the operation of time reversal becomes just the operation of complex conjugation of wave function. And if time reversal is present then complex conjugation does not alter the energy of stationary states and the stationary states are twofold (at least) degenerate. So, the complete wave function for the given energy is linear superposition of two mutually conjugate wave function, i.e. it is a REAL wave function. Now, we also heared from L&L that electric field does not remove time reversal symmetry (for stationary case read: invariance under complex conjugation of wave function). I hope this clarifies the question addressed by Dr. Herbert H. H. Homeier. Let me continue - I still feel I have more to say. When complex wave function is necessary ? The quantity which allows us to be in control over this is the QUANTUM-MECHANICAL CURRENT. It is a one line calculation to show that it identically cancels if wave function is real. See please A.S.Davydov "Quantum mechanics", (Pergamon, 1976, Int. Series in Nat. Phil. vol.1). On page 50, formula (15.9) he clearly explains the phenomenon of fundamental dependence of current on complex phase of wave function. Now, we see that all our scaterred knowledge on cases when COMPLEX wave function is necessary fits into this scheme: 1) When orbital momentum has definite value of its 3rd component then it is a circular current of electron around the nucleus (and the wave fucntion is e^(i*m*phi)); 2) When there is a plane wave (which is itself the pure complex phase) then it is a current of particles propagating in some direction, the momentum has definite values; 3) When there are periodic boundary conditions then we have Bloch functions -- the presence of microcurrents associated with electron spins makes them necessarily complex. Periodic motion is NOT that general criterion: while periodic circling of electron around the nucleus causes complex wave function (case 1), the motion of harmonic oscillator is perfectly described by real wave functions. I hope I made clear this point too. It is easy to speculate on the consequences of the fundamental dependence of quantum-mechanical current on the phase of the complex wavve function. For instance, the magnetic field is immediately associated with the quantum-mechanical current if the particle is charged -- and it IS the case in molecular quantum physics (quantum chemistry). That's why all the argumentation about importance of magnetic effects for complex wave functions (or vise versa) appears. Another point: open shell implies uncompensated spin, uncompensated spin implies uncompensated magnetic moment and therefore non-zero magnetic field, and, at last, it implies existing quantum-mechnical current associated with it. So, Dr. Herbert H. H. Homeier was right and we were able to put his remark into systematic approach - open shell necessarily requires complex wave function. Dr. Herbert H. H. Homeier made also few useful remark about technical details involved when complex functions are used. I think it is a separate important question whose status is equal to status of any general question about use of a computational method. It is, at present, more important to make people convinced that further essential progress in quality of calculations is impossible without complex wave functions and without including the associated magnetic terms into molecular Hamiltonian (spin-orbit, and afterwards hyperfine terms). Continuing use of Coulomb Hamiltonian only at the continuously increasing numerical accuracy is a dead end. The omitted magnetic terms are quite large on scale of accuracy claimed nowadays even by standard packages. It is time for chemists to give up quantum mechanics of thirties and turn to the modern quantum mechanics. It is time for physicists to understand that computer progress made available many challenging problems in molecular physics - the problems which were avoided as plague since they were not treatable by analytical techniques and were therefore "not fundamental". Crisis of high energy physics and high cost physics in general "helps" physicists to turn to molecular physics. Growth of computer might and attainable accuracy presses chemists to reconsider formalism employed in thirties and based on Coulomb interaction only. Sincerely, Ayaz Bakasov. From gavin@vangogh.chem.uab.edu Mon Apr 3 18:27:34 1995 Received: from vangogh.chem.uab.edu for gavin@vangogh.chem.uab.edu by www.ccl.net (8.6.10/930601.1506) id SAA24170; Mon, 3 Apr 1995 18:27:32 -0400 From: Received: by vangogh.chem.uab.edu (AIX 3.2/UCB 5.64/4.03) id AA17791; Mon, 3 Apr 1995 17:21:46 -0500 Date: Mon, 3 Apr 1995 17:21:46 -0500 Message-Id: <9504032221.AA17791@vangogh.chem.uab.edu> To: CHEMISTRY@ccl.net Subject: INSTABILITY ANALYSIS Cc: GAVIN@vangogh.chem.uab.edu Dear Nettlers, Someone can show me how to do the instability analysis ? ANY INFO will appreciate! Gavin Tsai From gavin@vangogh.chem.uab.edu Wed Apr 5 12:48:54 1995 Received: from vangogh.chem.uab.edu for gavin@vangogh.chem.uab.edu by www.ccl.net (8.6.10/930601.1506) id MAA03543; Wed, 5 Apr 1995 12:48:52 -0400 From: Received: by vangogh.chem.uab.edu (AIX 3.2/UCB 5.64/4.03) id AA09878; Wed, 5 Apr 1995 11:43:08 -0500 Date: Wed, 5 Apr 1995 11:43:08 -0500 Message-Id: <9504051643.AA09878@vangogh.chem.uab.edu> To: CHEMISTRY@ccl.net Subject: Instability Dear Netters, I studied Ab Initio calculations of NaOONO. There are some unusual results observed: (a). Unusual high O=N stretching in MP2/6-311+G* calculation. EXP. O=N: 1437 cm-1 in frozen argon. HF/6-311+G* O=N: 1718 cm-1 MP2/6-311+G* O=N: 2408 cm-1 -- unusual high! (b). Unusual IR intensity in O=N stretching. MP2/6-311+G* O=N: 55012 km/mol (IR intensity is 500 km/mol for instense band) (c). O=N stretching is very sensitive to different basis set. MP2/6-311+G* O=N: 2408 cm-1 [55012] km/mol -IR intensity MP2/6-311+G O=N: 1797 cm-1 [7432] MP2/6-311 G* O=N: 2416 cm-1 [55314] MP2/6-311 G O=N: 1829 cm-1 [7676] I think these unusual results are caused by the instability of HF wavefunctions. So, I calculate instability analysis in G92/DFT. I got this messages: " Unable to determine singlet-triplet character of expansion vector. Closed Shell can not continue. Error termination in Lnk1e. " The G92/DFT was able to determine what instability this wavefunction has, RHF -> UHF, but not the orbitals involved. We found that there is a triplet state that is lower in energy than the singlet at HF levels of theory, but singlet is lower for MP2 and DFT levels of theory. Triplet states were searched for, but all of those obtained from the INDO guess were high in energy. The lowest triplet was not found by the INDO guess. There are four references about instability: (1). J. Cizek and J. Paldus, J. Chem. Phys. 47 (1967) 3976. (2). J. Paldus and J. Cizek, Chem. Phys. Letters 3 (1967) 1. (3). J. Paldus and J. Cizek, J. Chem. Phys. 52 (1970) 2919. (4). J. Paldus and J. Cizek, J. Chem. Phys. 54 (1971) 2293. ANY INFO and advise will appreciate! Gavin Tsai From JWANG@ac.dal.ca Thu Apr 6 20:59:14 1995 Received: from SYSWRK.UCIS.Dal.Ca for JWANG@ac.dal.ca by www.ccl.net (8.6.10/930601.1506) id UAA10748; Thu, 6 Apr 1995 20:59:08 -0400 From: Received: from AC.Dal.Ca by SYSWRK.UCIS.DAL.CA (PMDF V4.3-13 #6307) id <01HP18IDZGS0003YCO@SYSWRK.UCIS.DAL.CA>; Thu, 06 Apr 1995 21:58:56 -0300 Received: from AC.DAL.CA by AC.DAL.CA (PMDF V4.3-13 #6307) id <01HP18H3AYM800515W@AC.DAL.CA>; Thu, 06 Apr 1995 21:58:45 -0300 Date: Thu, 06 Apr 1995 21:58:44 -0300 Subject: CCL:Summary of internal coordinates --> normal coordinates To: CHEMISTRY@ccl.net Message-id: <01HP18H3LERM00515W@AC.DAL.CA> Dear Netters, Many thanks to people who replied to my query about converting internal coordinates to the normal coordinates. Here is a summary of replies I have received. Jian Wang Department of Chemistry Dalhousie University Halifax, Nova Scotia Canada B3H 4J3 Tel: (902)-494-7921 Fax: (902)-494-1310 ---------- Original Question ---------- Dear Netters, I am looking for a code to calculate the matrix which converts the internal coordinates to the normal mode coordinates. I can get the force constant matrix with respect to the internal coordinates, but I need a code to calculate the G matrix, which is necessary for the diagonization of the FG matrix. From there I will be able to obtain the transformation matrix. Thanks in advance. Jian Wang ------------ Replies ------- From: IN%"cas@softshell.com" "Craig Shelley" ============================== I will be gone the week of March 20th. I will be back on the 27th. If you need support or other information before then, e-mail development@softshell.com, call our office at 303-242-7502, or fax at 303-242-6469. Thanks, Craig A Shelley From: IN%"youkha@biosym.com" ============================== If you already have code that calculates normal modes in cartesian X = Lx Q and to determine the "B matrix" R = B X ... (a program that does the former does the latter if the force constants are expresse d in internal space) then the Lr matrix you are looking for is Lr = B * Lx where R = int coords X cartesian coords Q normal coords this is much simpler. programs at QCPE exist for these things. see the book of Wilson on that topic ************************************************************************** Dr. Philippe Youkharibache e-mail: youkha@biosym.com Biosym Technologies Inc. 9685 Scranton Road tel: (619) 546 5562 San Diego, CA 92121 fax: (619) 458 0136 ************************************************************************** From: IN%"GOVENDEM@che.und.ac.za" "Magan Govender - PG" ============================== Dear Sir, I have some codes available, and shall be happy to forward them to you, on request. Greatings from South Africa M.G. Govender Centre for Theoretical and Computational Chemistry Dept of Chemistry University of Natal King George V Avenue Durban South Africa M.G. Govender's second reply: ============================== Dear Sir, The program we are using does a complete vibrational calc. of the secular equation, it is called Vibra by Dr. STeele. We also have ASYM20, by Ian Mills , obtained from QCPE. A also have a code from SPectro, author Handy, and Willets, I can send you this code, and lastly there is a program obtainable via ftp iqm.unicamp.br, in the chem or pub dir, which is called BGF and NCA, similar to ASyM20. Anyway I shall forward you the code soon which is in fortran. The address of Dr STeele is aslc801@vmsfe.ulcc.ac.uk. regards Magan From: IN%"nowak@ibm320.chemie.th-merseburg.de" ============================== Dear Jian Wang If you have the B matrix which transform the cartesian force constant matrix to the internal force constant matrix you can build the G matrix by G=B M**-1 B#. Where B# is the transpose of the B matrix and M**-1 is the inverse of the M matrix. M is a diagonal matrix of tripels of the mass. The product FG is a nonsymmetric matrix. The simplest way to get the eigenvalue and eigenvectors is to bring the matrix to a hessenberg matrix then you can get the eigenvalue and eigenvectors by the qr methode. I wrote a FORTRAN program which can build the internal force constants matrix from the cartesian force constant matrix. Futhermore it will calculate the G matrix and frequences. There is also a procedure which bild the B matrix. If you are interested in my program please send me a e-mail. If you are loocking for some literature for this topic: Califano Vibrational States JOHN WILLEY & SONS 1976 T.Miyazawa J. Chem. Phys. 1958 29 P.246 R.J.Malriot J. Chem. Phys. 1955 23 P.30 Futhermore I am intrested in your way to get the internal force constant matrix. Thanks in advance. Thomas Nowak e-mail nowak@ibm320.chemie.th-merseburg.de -------------- End of Replies ---------