From chemistry-request-!at!-server.ccl.net Fri Aug 25 12:51:24 2000 Received: from ivory.trentu.ca (trentu.ca [192.75.12.103]) by server.ccl.net (8.8.7/8.8.7) with ESMTP id MAA27046 for ; Fri, 25 Aug 2000 12:51:24 -0400 Received: from trentu.ca ([204.225.13.50]) by trentu.ca (PMDF V5.2-32 #29543) with ESMTP id <01JTDYVYHW62000FP6 %-% at %-% trentu.ca> for chemistry %-% at %-% ccl.net; Fri, 25 Aug 2000 12:51:07 EDT Date: Fri, 25 Aug 2000 12:55:45 -0400 From: elewars Subject: SUMMARY DFT ORBITALS, WF To: chemistry _-at-_)ccl.net Message-id: <39A6A510.163FFE7C;at;trentu.ca> MIME-version: 1.0 X-Mailer: Mozilla 4.7 [en] (WinNT; I) Content-type: text/plain; charset=us-ascii Content-transfer-encoding: 7bit X-Accept-Language: en Friday, 2000 Aug 25 Hello, here is the summary of replies to my question about orbitals and wavefunctions in DFT. I got 12 replies: #1 D. Mathieu, #2 and #3 E. Scerri, #4 E. Proynov, #5 H. van Dam, #6 A. Bittner, #7 J. Spanget-Larsen, #8 U. Richter, #9 A. Shusterman, #10 K. Nguyen, #11 J. Smith, #12 R. Jensen. I hope no one has been overlooked. Thanks very much to all who responded. I don't think question 3 can really be answered definitively yet. I'll leave this topic with the thought that an electron density is an observable (it can be measured), while a wavefunction is a recipe, a prescription, a Vorschrift--use it in a certain way and you get the right answers. THE QUESTION Sat 2000 Aug 19 Hello, We know that in pure DFT (in contrast to wavefunction theory) there are are no orbitals and there is no wavefunction; the Kohn-Sham orbitals of current practical DFT methods were introduced to make the calculation of the electron density tractable. QUESTIONS (1) An important feature of ab initio (and semiempirical) calculations is that one can visualize the orbitals, particularly the HOMO and LUMO (frontier orbitals) and make chemical deductions. In a _pure_ DFT calculation a question about the shape and energy of these orbitals would be meaningless, right? (2) The Fukui function (R. G. Parr and W. Yang, J Am Chem Soc, 1984, 106, 4049) seems to be always discussed in the context of DFT; yet this function is "a chemical reactivity index in the sense of the frontier-electron theory of reactivity..." (R. G. Parr and W. Yang, "Density-Functional Theory of Atoms and Molecules", Oxford, New York, 1989, p. 101). But in pure DFT theory there are no frontier orbitals. Isn't this strange? (3) If, as has been suggested (Parr and Yang, "Density-Funtional Theory...", p. 53) wave mechanics can be reformulated without the wavefunction concept, what are we to make of the 80-year-old debate in physics and philosophy about the meaning of the wavefunction (wavefunction collapse, Schroedinger's cat, the many-worlds theory of QM, etc, etc, etc; see e.g. "Einstein, Bohr and the Quantum Dilemma", A. Whitaker, Cambridge University Press, Cambridge, 1996, and _many_ other books and papers...). Was this debate a meaningless exercise that would never have occurred if QM had been originally formulated in terms of electron (or more generally, particle) density rather than the wavefunction? Thanks, E. Lewars ==== THE REPLIES #1 Didier Mathieu, mathieu.,at,.ripault.cea.fr > In a _pure_ DFT > calculation a question about the shape and energy of these orbitals > would be meaningless, right? While the concept of orbitals is not necessary, it might be possible to define orbitals from the density obtained from "pure DFT"and its responses to changes of the Hamiltonian. It might then turn out that such orbitals, wile no physical observables, can be given some physical meaning a posteriori. > 1989, p. 101). But in pure DFT theory there are no frontier orbitals. > Isn't this strange? Why ? Fukui functions are defined independently of the notion of orbitals. Thus, I see them as an natural alternative - whithin DFT - to frontier orbitals and perturbation theory in order to interpret chemical reactivity. > papers...). Was this debate a meaningless exercise that would never have > occurred if QM had been originally formulated in terms of electron (or > more generally, particle) density rather than the wavefunction? In that case, I wonder how these physical concepts would have emerged. Clearly, the one-particule density is not suitable as it does not contains all the information about the system. However, QM has also been formulated in terms of the density matrix since the beginning. For me, probably for historical reasons, it is easier to grasp many QM ideas in terms of the wavefunction, also because some of the basic notions involve "interference effects", (sorry for being so conditioned by the wavefunction picture) and the phase of the wavefunction might be more suitable than the coherences of the density matrix to study them. On the other hand, in some cases, the influence of the environment on the quantum system considered is more naturally introduced in the density matrix formulation: for instance when adding the phenomenological relaxation rates T1 and T2 to the evolution equation in NMR. This sounds more natural than modifying the Shrodinger equation through adding a non-hermitian term to the hamiltonian. Cheers, Didier. === #2 Eric Scerri, scerri -x- at -x- purdue.edu But surely the shape of the electron density which would correspond to these orbitals could still be visualized if necessary, just as electron density can be in the ab initio wavefunction approaches. The two approaches are alternative ways of looking at the micro world. Whether one chooses to visualize, or not, is always an option in any theory. But perhaps you ar asking a more specific question like what do HOMO and LUMO electron density plots look like? Please clarify your question (1) eric scerri ------------------------------------------------- Eric Scerri PhD, Visiting Professor, Department of Chemistry & Biochemistry, UCLA, Los Angeles, CA 90095 USA --- I replied: Hello, I'll summarize the answers, maybe with some comments. But concerning your answer: in wavefunction theory we have a series of atomic or molecular orbitals, psi_1, psi_2,..., with increasing energy. I think that in a pure density functional theory we just have an electron density function, rho(r); not a series of electron densities. So there is no electron density rho_1 corresponding to a certain orbital , and another density rho_2 corresponding to another orbital, etc. Just one density function. But maybe I have overlooked something--so I am asking the chemical community for their thoughts. And it just occurrs to me that one might wonder how this fits in with photoelectron spectroscopy, which indicates that molecules have groups of electrons of different energies. thanks EL ==== #3 Eric Scerri I dont know enough about DFT to know the answer to this further point. I would guess that it is possible to recover some kind of shell tructure from DFT. Please let me know what you discover from others. I am also still puzzling over your No 3. You ask interesting philosophical questions. Please consider joining the Philchem list. the address can be found on the web pages for the International Society for the Philosophy of Chemistry after the section on the recent meeting in Poznan. ==== #4 Emil Proynov, proynov \\at// chimie.umontreal.ca Dear Dr Elewards, I would suggest not to get so upset by those who claim that the Kohn-Sham (KS) determinantal wavefunction does not have a physical meaning. The SCF potential, as defined in the KS DFT scheme, determines a self-consistent field such that the KS determinant minimizes the electron kinetic energy and yields the exact electron density. It differs from the Hartree-Fock (HF) self-consistent field in what the latter is associated with the HF determinant (minimizes the total energy but does not yield the exact electron density). To any meaningful self-consistent field one can assign SCF one electron orbital picture with a portion of physical/chemical meaning in it. To my opinion there is nothing mysterious in the KS SCF field, just to keep in mind that it does not originate directly from the Schrodinger equation but from the Euler equation of the DFT variational task. It is true that in the latter one does not deal directly wavefunctions, but all the quantum mechanics with its postulates, theorems and paradoxes is hidden there in the exchange-correlation potential. You can find different opinions on this in the literature indeed. The matter is perhaps more complicated that I am trying to make it with this comment. Just to note that an increasing amount of theoretical studies use and refer explicitly to the KS SCF orbitals demonstrating their practical utility (see for example: I. Verdernikova et.al. Int. J. Quant. Chem. vol.77, 161 (2000)). Best wishes, Emil Proynov +-----------------------------+ | Emil Proynov | | proynov;at;chimie.umontreal.ca | +-----------------------------+ ===== #5 Hub van Dam, h.j.j.vandam&$at$&dl.ac.uk (Question 1) People try that sort of thing, yes. But I am not sure if this really works in cases where there is strong static correlation. In other words where you have a wavefunction (again) with multi configurations with large coefficients. (Q2) Yes, this seems very strange. (Q3) The 74 year old debate, autumn next year the Schroedinger equation will have been around for 75 years (I haven't heard any plans yet of how people want to commemorate this anniversary). I would like to think that quantum mechanics only obtained a firm foundation with Schroedingers wave-equations. Anyway, I don't think this was a meaningless exercise. Although the wavefunction in itself has no direct physical interpretation, we still need it to get the probability density right. I.e. the probability to find a particle at a certain point in space behaves like there is some superposition of waves involved. Trying to formulate quantum mechanics in terms of the electron density only, and at the same time including these superposition effects correctly is a far from trivial matter (I think it is save to say that this problem is unsolved). Therefore I think that although the wavefunction has not direct physical interpretation it is far from "meaningless" concept. Kind regards, Huub -- ======================================================================== Huub van Dam E-mail: h.j.j.vandam- at -dl.ac.uk CCLRC Daresbury Laboratory phone: +44-1925-603362 Daresbury, Warrington fax: +44-1925-603634 Cheshire, UK WA4 4AD ====== #6 Eric Bittner, bittner.,at,.uh.edu No, There is the issue of measurement, interference, causality, etc.. which you can not simply sweep under the rug by invoking a density. The debate over the ontology of quantum mechanics continues and is perhaps extremely relevant in the advent of quantum computers. Perhaps if your only interested in using quantum theory as an algorithm for making predictions about physical reality, the debate is pointless. However, if you really and truly want to understand the inner workings of nature, it is not. IF you do choose to only work with the density, rho(x), you are not representing the full physical picture. QM is completely formulated in terms of the density, which evolves under the continuity equation, AND the quantum Hamilton-Jacobi equation, \frac{\partial S}{\partial t} - H(p,q) - \frac{\hbar^2}{2m}\frac{1}{\sqrt{\rho}}\nabla^2 \sqrt{rho} = 0 where H(p,q) is the Hamiltonian with the canonical substitution p-> \nabla S/m and the last term is the quantum potential, Q. If you seek stationary states, i.e. d\rho/dt = 0, then the QHJ reduces to the time-independent Schrodinger equation. These equations of motion are easily derived from the Schrodinger eqn. by writing psi = R exp(i S/hbar) and defining the density as rho = |psi|^2. On the computational side of this: Bob Wyatt and I have recently developed methods to solve these equations of motion using ONLY the trajectories they imply...no wave function, no basis functions...only quantum paths. A pre-print version of one of our papers is included as an attachment (pdf format). -Cheers, Eric R. Bittner U. Houston -- Prof. Eric R. Bittner Department of Chemistry Univ. of Houston ========== #7 Jens Spanget-Larsen, jsl.,at,.virgil.ruc.dk (Q1) Or, in other words: "What Do the Kohn-Sham Orbitals and Eigenvalues Mean?". Stowasser and Hoffmann address this question in their recent paper - Ralf Stowasser & Roald Hoffmann: J. Am. Chem. Soc. 121, 3414-3420 (1999). Yours, Jens >--< =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= JENS SPANGET-LARSEN Phone: +45 4674 2000 (RUC) Department of Chemistry +45 4674 2710 (direct) Roskilde University (RUC) Fax: +45 4674 3011 P.O.Box 260 E-Mail: JSL*- at -*virgil.ruc.dk DK-4000 Roskilde, Denmark ========= #8 Uwe Richter, uwe |-at-| nist.gov Hi, I'm very interested in answers to this questions. Please could you summarize or send me the responses directly? There are some people who even discuss virtual KS orbitals. Thanks, Uwe =========== #9 Alan Shusterman, Alan.Shusterman /at\directory.reed.edu These are great questions. I'd be grateful if you would pass along any answers you receive. Thanks, -Alan alan-: at :-reed.edu ==== Alan Shusterman Department of Chemistry Reed College Portland, OR ========= #10 Kiet Nguyen, Kiet.Nguyen # - at - # wpafb.af.mil Although there is considerable controversy in the literature concerning the meaning of Kohn-Sham orbitals, I think they are far from "meaningless". This topic has been carefully analyzed and reviewed.[1-5] Parr and Yang stated that "... one should expect no simple physical meaning for the Kohn-Sham orbital energies."[3] However, using the Janak theorem which "provides a meaning for the eigenvalues of the Kohn-Sham equation", they have connected the HOMO and LUMO energies to the ionization potential and electron affinity, respectively.[3] [1] E. J. Baerends and O. V. Gritsenko, J. Phys. Chem. A 101 (1997) 5383. [2] R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (University Press, Oxford, 1989). [3] J. P. Perdew, in: Density Functional Methods in Physics, edited by R. M. Dreizler and J. da Providencia New York, 1985), p. 265 [4] E. J. Baerends and O. V. Gritsenko, J. Phys. Chem. A 101 (1997) 5383. [5] R. Stowasser and R. Hoffman, J. Am. Chem. Soc. 121 (1999), 3414. Kiet A. Nguyen AFRL/MLPJ Laser Hardened Materials Branch Wright-Patterson AFB, OH 45433 Phone (937) 255-6671, Ext 3178 FAX (937) 255-1128 ========== #11 Jack Smith, Smithja _-at-_)ucarb.com (We know that in pure DFT...no wavefunction) At least not explicitly. One-electron orbitals are not required for wavefunction theory either. I would say that the two approaches converge when you reduce it to N-electron density matrices (probably even at the 2-electron density matrix level). (The KS orbitals...electron density tractable) Particularly to make the kinetic energy part tractable. Orbitals also made wavefunction theory more tractable by introducing the pseudo-one-particle SCF equations, which further allowed for the formulation of the Roothan equations (for a basis set expansion approach). > My feeling is that Natural Orbitals from a CI calculation (eigenvectors of > 1-particle density matrix) should correspond to KS orbitals with > fractional occupations. Is there an analog to Natural Geminals (from the > 2-particle density matrix) in DFT? (In a pure DFT calc., a question about the shape and energy of these orbitals would be meaningless, right?) Orbitals, yes; orbital densities, no. The density differences between the neutral system and its various ionic states are in principle observable quantities (assuming you can get xray diffraction data for the ions). Under Koopmans' approximation these density differences should be the same as the orbital-derived densities. And the orbital energies should correspond to the ionization (electron attachment) energies. A similar correspondence can be defined for orbital density differences among different excited states. One thing that's missing from a pure density picture is the concept of overlap and orthogonality, which depend on the sign of the underlying one-electron orbitals (wavefunctions). Overlap is a key component of frontier orbital theory. Should it be? Is there a pure density analog? (The Fukui function.. But in a pure DFT theory these are no orbitals. Isn't this strange?) But there ARE frontier orbital DENSITIES. The +/- Fukui functions should correspond to the orbital densities mentioned above for whole electron changes (N +/-n). However, in an "open" sysem (in a Grand Canonical sense) where fractional electrons can come and go, the Fukui function can also be defined for infinitessimal changes in electron number - a continuous functional of the number of electrons. Thus, derivatives in the density wrt electron number can be defined even at N electrons. The corresponding derivative in energy would be the chemical potential, and the second derivative would be the hardness -- with no need to define a HOMO or LUMO and use crude finite difference approximations (along with Koopmans' approximation). > (3) > If, as has been suggested (Parr and Yang, "Density-Funtional Theory...", > p. 53) wave mechanics can be reformulated without the wavefunction > concept, what are we to make of the 80-year-old debate in physics and > philosophy about the meaning of the wavefunction (wavefunction collapse, > Schroedinger's cat, the many-worlds theory of QM, etc, etc, etc; see > e.g. "Einstein, Bohr and the Quantum Dilemma", A. Whitaker, Cambridge > University Press, Cambridge, 1996, and _many_ other books and > papers...). Was this debate a meaningless exercise that would never have > occurred if QM had been originally formulated in terms of electron (or > more generally, particle) density rather than the wavefunction? > It just means that all the debate is focused on the form of energy functional itself, the exchange and correlation parts in particular, rather than the wavefunction. One still has to explain the particle-wave duality of the electron beam experiments. The outcome is still probabilistic (non-deterministic for a given electron) whether you speak in terms of electron density or a wavefunction. The concepts of exchange and correlation are still very much "quantum" phenomena, whether you encapsulate in a quantum-constrained probabilistic wavefunction and a classical Hamiltonian (energy) operator, or in a classical-like probabilistic electron density and a quantum-constrained energy functional. DFT-KS and HF only converge when HF exchange is used, no correlation term is included, AND the Grand Canonical (GC) form of the density operator is used. DFT-KS and MC-SCF converge as the MC-SCF approach a full CI and the correlation functional approaches the "exact" expression, but again only if the GC form of the density operator is used. An interesting middle ground would be a Natural Orbital iteration on a small CI, or a VB wavefunction using non-orthogonal orbitals, or GC UHF using localized fractionally occupied nearly-orthogonal orbitals. (As you might tell, I'm a an advocate of the latter - which should have a correspondence with DFT-KS, given the right functional). My 2 cents' worth. - Jack -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- Jack A. Smith || Union Carbide || Phone: (304) 747-5797 Catalyst Skill Center || FAX: (304) 747-4672 P.O. Box 8361 || S. Charleston, WV 25303 || smithja%!at!%ucarb.com -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- ===== #12 Roy Jensen, royj |-at-| uvic.ca Have you had any response to your questions. I would be very interested to hear the results. Please post a summary to the CCL. Roy Jensen =================== ================ =========