Research: Science and Education
Physical Interpretation of Koopmans’ Theorem: A Criticism of the Current Didactic Presentation Celestino Angeli Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56100 Pisa, Italy
Koopmans’ theorem (1) (KT) is part of practically all the preliminary courses and textbooks of molecular quantum chemistry. The reasons for such wide consideration can be summarized as follows:
determinant wave function | Ψ 0典, built up on a set of N orthonormal spin orbitals {χ1, χ2, …, χ N }:
1. Its demonstration needs to use only concepts introduced to develop the Hartree–Fock (HF) approximation. 2. It gives the simplest approximation to evaluate ionization potentials (IPs). 3. It is derived from a simple physical assumption about the ionization process. 4. It allows one to attribute a physical interpretation to the eigenvalues of the Fock matrix and to justify the name “orbital energies” for them.
The HF approximation consists in finding those spin orbitals which minimize the mean value (E0) of the electrostatic Hamiltonian (Ᏼ) on |Ψ 0典.
These four points clearly show the didactic importance of KT as an application of the HF approximation. Regarding the practical use of KT in research, we note that through the years some improvements and extensions have been presented, as in the case of perturbation KT (2, 3) and extended KT (4–6), allowing KT to be considered as a starting point in a scheme of methods with increasing accuracy. Although not very accurate, KT is still widely used when a qualitative interpretation of the ionization process is needed and more refined methods are impracticable. In spite of such a widespread interest in KT, it is not easy to find in the literature a clear statement about the role played in its demonstration by the canonical HF orbitals and by the eigenvalues of the Fock matrix. These difficulties are due to the fact that the approaches followed in some widely used textbooks do not consider (7–10) or do not completely clarify (11) some basic details which are necessary in order to correctly interpret KT (especially to justify the term “orbital energies”). Such a situation is found also in specialized papers, with some sporadic exceptions (in addition to the original Koopmans paper, see for instance refs. 4, 12–14). Therefore we find it worthwhile to reconsider the didactic scheme used to present KT. In the next section we shall give a summary of the present approach to KT together with a criticism of it. In the last section we shall propose a new way to present KT, which explicitly takes into account some commonly overlooked details and which seems to us interesting from a didactic point of view because it is based on the concept of constrained optimization. Today’s Presentation of Koopmans’ Theorem In this section we give only a brief summary of the HF approximation and the derivation of KT as they are usually proposed in elementary courses in quantum chemistry. We shall use the notation of the textbook Modern Quantum Chemistry by Szabo and Ostlund (9, pp 123–128). Let us assume that the chemical system under consideration (for the sake of simplicity supposed to be a closed-shell system with N electrons) can be described as a single Slater 1494
| Ψ 0典 = | χ1, χ2, …, χ N 典
N
(1)
N N
E 0 = Ψ0 Ᏼ Ψ0 = Σ i ᐈ i + 1 Σ Σ 2 i=1 j=1 i=1
i i | j j – i j |i j
(2)
with
i ᐈ j = χ i ᐈ χ j = 兰d x1 χ 1* x1 ᐈ x1 χj x1
(3)
where h(x1) is the usual one-electron core-Hamiltonian, and
i j |k l = χi χj |χk χl =
兰d x1 d x2 χ i * x1 χj x1
1 χ*x χ x r 12 k 2 l 2
(4)
The minimization condition on E0 leads to the requirement that the occupied spin orbitals {χ1, χ 2, …, χN } span an invariant subspace of the Fock operator f N
f 冏 χa典 = Σ εab 冏 χb 典, ∀a ∈ 1, 2, …, N b=1
(5)
where f is defined as N
χ i f χ j = χ i ᐈ χ j + Σ i j | k k – i k |k j k=1
(6)
Equation 5 expresses the requirement that the matrix elements of f between occupied and virtual spin orbitals must be zero. At this point we note that there is not a unique set of HF spin orbitals. Indeed, a unitary transformation of a set of N spin orbitals satisfying eq 5 produces a new set of spin orbitals which also satisfy the same equation (that is | Ψ 0 典 and E0 are invariant under a unitary transformation of the doubly occupied orbitals). In particular, we can choose the unitary transformation in such a way that the matrix ε with elements
εij = χ i f χ j
(7)
will be diagonal. These spin orbitals are called “canonical spin orbitals”. We now evaluate the IP of the system, that is, the energy required to take away an electron from the neutral molecule. It is clear that if we want to approximate the IP, a good description (albeit approximated) of both the neutral and the ionic molecule is required. We describe the neutral molecule
Journal of Chemical Education • Vol. 75 No. 11 November 1998 • JChemEd.chem.wisc.edu
Research: Science & Education
at the HF level. One can assume that the electronic density of the ionic system is roughly described as the one of the neutral system minus the density of the orbital from which the electron has been taken away. This amounts to supposing that the other N – 1 electrons do not change their spatial distribution during the ionization process (frozen orbital approximation). Therefore we describe the ionic molecule with a single determinant by using the HF spin orbitals of the neutral molecule. With these assumptions Koopmans’ theorem is usually enunciated as follows: the energy difference between the HF energy E0 of an N-electron system and the mean value of Ᏼ over an N – 1-electron single determinant built up using the HF canonical spin orbitals of the N-electron system is equal to the orbital energy relative to the missed canonical spin orbital, and it is an approximate estimate of the ionization potential of the N-electron system. A similar theorem can be enunciated for the electron affinity, but in this paper we shall not deal with this second formulation of Koopmans’ theorem. The demonstration of the theorem is usually limited to show that if |(N–1) Ψ c 〉 = |χ 1 χ 2,…,χ c–1 χ c+1 ,…,χ N〉
(8)
is the (N – 1)-electron single determinant describing the ionized system (χ i are the canonical HF spin orbitals of the neutral system) then (N–1)E – NE = 〈(N–1)Ψ |Ᏼ| (N–1)Ψ 〉 – 〈 NΨ |Ᏼ|N Ψ 〉 = ε ᎑ cc c 0 c c 0 0
we propose an alternative approach, which seems to us complete and appealing from a didactic point of view. An Alternative Approach to Koopmans’ Theorem As already stated, to evaluate the first IP we need to calculate the energies of the ground states of the N-electron and (N – 1)-electron systems. Our aim is to obtain a good estimate of the energy difference (the IP) by performing an approximate calculation such as HF, rather than the best absolute energies of the two systems within the same approximate method. Let us first consider the best IP value once the atomic basis set has been chosen, that is, the energy difference between two full configuration interaction (FCI) results: E FCI – NE FCI = (N–1)EHF – NE HF + (N–1)Ecorr – NEcorr (10)
(N–1)
where E FCI and EHF are the FCI and HF energies and Ecorr is the correlation energy defined as EFCI – E HF. Suppose we have calculated the HF energy of the neutral molecule NE HF. We now look for a simple method to estimate the quantity (N–1)Eeff = (N–1)E HF + ((N–1)E corr – NEcorr ), which we attribute artificially to the ion. The variational principle ensures that the inequality Ecorr ≤ 0 is always verified. Moreover, in general |(N–1)Ecorr| < |NEcorr| because the neutral system, having one electron more than the ionic one, is supposed to have a larger correlation energy. Therefore the difference (N–1) Ecorr – NEcorr is usually greater than zero and thus (N–1)E
(9)
This relation can be easily obtained from eq 2 by isolating the terms corresponding to i,j = c. Actually, eq 9 is nothing but a general relation between the energies of single determinants and it has no physical content. It is valid for a set of N orthonormal spin orbitals however chosen: neither the condition for ε to be diagonal, nor even the less restrictive condition of eq 5, needs to be satisfied. If the canonical spin orbitals are used, the matrix element εcc is an eigenvalue of the Fock matrix. An ambiguity is present at this point in the choice among the various HF spin orbital sets, because they are totally equivalent for the neutral system. The importance of Koopmans’ contribution was essentially in showing that the “canonical spin orbitals” are the best HF spin orbitals because they give the lowest energy for the ion. Therefore the IPs are approximated by the eigenvalues of the Fock matrix. This consideration was clearly expressed in Koopmans’ original paper (1) and several authors later pointed it out in primary chemistry research journals (see for instance refs 4, 12–14 ). However, less attention has been devoted to clearly explaining these fundamental points in textbooks of molecular quantum mechanics. The problem of choosing between the different sets of HF spin orbitals is to our knowledge faced only in McWeeny’s book (11), where a quite synthetic explanation is given: “This [relation IP = ᎑εcc ] is a legitimate approximation to IP only when the spin orbitals satisfy the canonical HF equation, for then NE0, εcc , and consequently the difference (N–1)Ec are all stationary against spin orbitals variation.” (We have changed some symbols from the original citation to match our formalism.) We deem further elaboration of the reason for choosing the canonical spin orbitals worthwhile. In the next section
eff
>
(N–1) E
(11)
HF
A schematic representation of the diagram of the energy considered here is reported in Figure 1. A way of calculating an energy of the ion that satisfies this condition (and which we suppose to be an estimate of (N–1) Eeff ) is to perform an HF calculation in a subspace S of the space spanned by the atomic basis set used for the neutral molecule. We shall call this energy “partial HF energy”, (N–1) Epart . In order to identify the S space we can take up again the frozen orbital approximation. Within this approximation, S is the space spanned by the occupied HF spin orbitals of
N–electron system
(N – 1)–electron system
FCI IP
N–1E
eff
N–1E
HF
N–1E
FCI
HF IP NE
HF
FCI IP
NE
FCI
Figure 1. Energy level diagram for neutral (N -electron) and ionic ([N – 1]-electron) systems. For the meaning of the energies NE FCI, (N ᎑1)E N (N ᎑1)E , and (N ᎑1)E FCI , E HF, HF eff see text.
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the neutral molecule. (This space is univocally defined and it does not depend on the nature of the HF spin orbitals, canonical or not.) The HF variational condition over the S space for the (N – 1)-electron system requires finding a set of N – 1 spin orbitals {χ′1, χ′2, …, χ′ N–1} that minimize the energy (N–1)
|
Epart = 〈 (N–1)Ψ1|Ᏼ| (N–1)Ψ1 〉
(12)
Ψ 1〉 = | χ′1, χ′2,…, χ′(N–1) 〉
(13)
(N–1)
The subscript 1 indicates that we are considering the first IP. The spin orbitals {χ′1, χ′2, …, χ′(N–1)} are obtained by taking away the N th one from the basis set {χ′1, χ′2, …, χ′N } originated by a unitary transformation of a set of HF spin orbitals of the neutral molecule {χ′1, χ′2, …, χ′N } = {χ 1 , χ 2, …, χ N }T
(14)
where T is a unitary matrix. We assume that the Nth spin orbital is chosen with α spin. From eq 9, we know that (N–1)
Epart = NEHF – ε′NN
(15)
where ε′ is the Fock matrix of the neutral molecule on the new basis set. It is clear that (N–1)Epart reaches its minimum value when ε′NN is maximized, that is, if ε′NN is the highest eigenvalue of the Fock matrix. Therefore, to satisfy the variational principle for (N–1)Epart , χ′N must be the canonical spin orbital related to the highest eigenvalue: χ′N ≡ χN. There are no conditions that allow us to specify the other N – 1 spin orbitals; that is, (N–1)Epart is invariant under unitary transformations of the basis {χ′1, χ′2, …, χ′(N–1)}. We note that the optimization reduces in practice to the optimization of the hole (the missing spin orbital) within the α subset of the spin orbital set, while the β subset remains unchanged. By taking away a β electron the derivation leads to the same value for the IP, which is an obvious result, given the double degeneracy of the doublet state of the ion. As for the second IP, a similar condition (taking away again an α electron), together with the constraint for | Ψ 2〉 to be orthogonal to | Ψ 1〉, leads us to identify the second eigenvalue of the Fock matrix as the approximate second IP and the second α canonical spin orbital as the spin orbital from which the electron has been taken away. In this way all the eigenvalues of the Fock matrix are shown to be approximations of the IPs. The quality of the estimate of the energy (N–1)Eeff within KT is impossible to evaluate a priori; therefore there are no guarantees that (N–1)E part approaches (N–1)E eff . However, the ordering of the IPs is usually correctly envisaged (for an exception see ref 15) and the errors of the calculated Koopmans IPs with respect to the experimental values are usually of the order of 10 %. At this point we can justify the use of the term “orbital energies”. We note that the usual argument is related to eq 9. It is therefore valid for any set of HF spin orbitals and it is not a prerequisite of the canonical one. On the other hand a complete demonstration of Koopmans’ theorem attributes a peculiar importance to the canonical spin orbitals because they allow an optimized description of the ion within the frozen orbital approximation.
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Conclusions In this paper we have revised the didactic approach to the KT, the demonstration of which is usually limited to verifying eq 9, which is misleadingly referred to as the theorem itself. The justification of the use of the canonical spin orbitals in that equation (and thus of the approximation of the IPs with the orbital energies) needs a further step. Without this step there are no reasons to prefer the canonical spin orbitals, and it is thus impossible to justify the particular role of the eigenvalues of the Fock matrix with respect to its diagonal elements in other representations (such as localized orbitals). To introduce these considerations in the demonstration of KT, we proposed a new approach, which starts from the consideration that an estimate of IPs using the HF method needs a partial optimization of the ion wave function if a complete optimization of the neutral wave function is performed. This allows the electron correlation error to be partially compensated. We note that in the usual approach, compensation of the two sources of error (orbital relaxation and difference in electron correlation) is presented as a fortuitous feature of KT, whereas our approach starts by introducing it as a specific requirement. The new approach introduces the choice between the different sets of HF spin orbitals of the neutral molecule as a usual problem of minimization, but on a subspace of the space spanned by the atomic basis set, leading to the conclusion that the best spin orbitals in this sense are the canonical ones. This approach allows us to present an example of partial optimization of wave function, which follows the study of the full HF optimization during a course of quantum chemistry. We note that the importance of Koopmans’ theorem is not related only to the study of ionization processes; it justifies the importance we attach to the canonical spin orbitals through the physical interpretation of the “orbital energies”, that is, the eigenvalues of the Fock matrix. The concept of orbital energies is of great importance for a qualitative understanding of a variety of chemical processes and it is commonly used by organic and inorganic chemists as well as theoretical chemists. For instance, it is used in the aufbau principle, a cornerstone for the first approach to atomic and molecular electronic structure even in undergraduate programs. An example of a careful use of the concept of orbital energies in the aufbau principle can be found in a recent paper in this Journal (16 ). However in this paper the quotation of KT is once more only partially correct: “In SCF theory an orbital energy εi corresponds to the energy that an atom would lose if an orbital Φi lost an electron while all the other orbitals remained ‘frozen’. This result is known as Koopmans’ theorem and suggests that the quantity ᎑ εi will approximate to an ionization energy”. Therefore as fundamental a concept as the one of “orbital energies” needs a clear explanation in a preliminary course in quantum chemistry. Acknowledgments I wish to thank R. Cimiraglia (University of Ferrara) and M. Persico (University of Pisa) for stimulating a discussion about the physical interpretation of Koopmans’ theorem. Literature Cited 1. Koopmans, T. Physica 1933, 1, 104.
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Information • Textbooks • Media • Resources 2. Decleva, P.; Lisini, A. Chem. Phys. 1985, 97, 95. 3. Chong, D. P.; Herring, F. G.; McWilliams, D. J. Chem. Phys. 1974, 61, 78. 4. Smith, D. W.; Day, O. W. J. Chem. Phys. 1975, 62, 113. 5. Day, O. W.; Smith, D. W.; Morrison, R. C. J. Chem. Phys. 1975, 62, 115. 6. Morrel, M. M.; Parr, R. G.; Levy, M. J. Chem. Phys. 1975, 62, 549. 7. Merzbacher, E. Quantum Mechanics; Wiley: New York, 1970. 8. Jørgensen, P.; Simons, J. Second Quantization-Based Methods in Quantum Chemistry; Academic: London, 1981.
9. Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry; McGrawHill: New York, 1989. 10. Almlöf, J. Lecture Notes in Quantum Chem. 1994, 64, 1. 11. McWeeny, R. Methods of Molecular Quantum Mechanics; Academic: London, 1989. 12. Nesbet, R. K. Adv. Chem. Phys. 1965, 9, 321. 13. Newton, M.D. J. Chem. Phys. 1968, 48, 2825. 14. Berthier, G. Stud. Phys. Theor. Chem. 1989, 62, 91. 15. Duke, B. J.; O’Leary, B. J. Chem. Educ. 1995, 72, 501. 16. Melrose, M. P.; Scerri, E. R. J. Chem. Educ. 1996, 499, 73.
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