Configuration-interaction

In configuration interaction (Cl), one mixes states arising from different spin-orbital configurations. In theory, one may expand the exact solution to Schrodinger s electronic equation (equation 5) in terms of the complete (infinite) set of determi-nantal wavefunctions, which in turn are constructed from some complete set of one-electron spin orbitals.33 This is obviously not a practical solution, and one must find instead, a smaller number of determinantal functions formed from the complete set which will give a close approximation to the true solution. 

A Cl state function Pi is expressed as a linear combination of the determinantal wavefunctions 0) formed from configurations of identical symmetry and spin, 

A restricted Hartree-Fock calculation on a closed-shell n-electron system using a basis set of N orbitals will produce n/2 doubly-occupied molecular orbitals and N—nj2 vacant or virtual orbitals. In a standard Cl calculation, the excited-state determinants are formed by systematically promoting electrons from the occupied orbitals of the ground-state determinant to the vacant or virtual orbitals. The number of configurations which can be formed in this way from N electrons and n basis functions178 is of the order of nN. Thus, even with today s high speed computers, a full Cl is possible only for very small systems. 

Minimal Configuration Interaction.—A minimum requirement of a potential energy surface is that the wavefunction continuously describe the system as the reaction takes place, producing reactants and products in their correct states. In order for this situation to be obtained, the correlation error must either be taken fully into account or at the very least, be approximately constant over that portion of the potential energy surface under consideration. As discussed and illustrated in the previous section, one should at least perform a limited configuration interaction in which those configurations needed to provide the correct description of the dissociation products are included. 

Truncated Cl Wavefunctions.—Between the limits of a minimal and a full configuration interaction one is faced with the problem of choosing how many configurations and more importantly, which configurations are to be included. A computational problem arises because of the very slow convergence, generally 

As an example of constructing a multiconfiguration wavefunction, consider two electron configurations and which are orthonormal, that is. 

Assuming that the interaction energy between and is nonzero, 

Then the state energies f (/= I. 2) are determined by solving the secular equation 

Similarly, (4 g H 3 2) = 0. Thus, the singlet excited configuration 4 s (equation 8.46) cannot mix into g. The only configuration of 8.3 that can mix with 4 g is the doubly excited configuration 4 e, since 

Without loss of generality, it may be assumed that Ef. Then if the energy difference between I g and 1 e is substantially greater than the interaction energy K 2 between them, we obtain the following results 

5 Theory of atomic bound states 5.6 Configuration interaction 

The coordinate—spin representation of the states n/jm) of an IV-electron atom or ion can be expanded in an M-dimensional linear combination of single-determinant configurations pk) (5.1). This is the configuration-interaction expansion. The orbitals a) forming the determinants are represented as orthonormal square-integrable functions f aix). 

We denote the atomic eigenstate n jm) by i) and the configuration-interaction expansion by 

The configuration-interaction approximation to i) results from diagonal-ising the atomic Hamiltonian H in the M-dimensional basis Ip ). 

If the states i ) and i) belong to different symmetry manifolds, characterised by the quantum numbers, j, then the Hamiltonian matrix element is zero. It is economical to consider the diagonalisation in a particular symmetry manifold and we will begin our discussion in this way. The basis states fk) are now symmetry configurations consisting of linear combinations of configurations which have the symmetry /j of the manifold. 

The total wave function can be much improved if it is written with the help of two or many configurations, each corresponding to a Slater determinant wave function. The most general way of writing a wave function of this type is 

In spin orbital i has been replaced by spin orbital a in spin orbitals i and j are replaced by spin orbitals a and b, etc. 

Another way to derive Equation 1.94 is to make a straightforward expansion of the total N-electron wave function. Let Dj be expansion coefficients  

If the antisynunetrizer is applied to Eqnation 1.95, we obtain an expansion of the same type as in Eqnation 1.94. 

In the following examples, we will show that configuration interaction (Cl) solves the correlation pioblan, in principle. Unfortunately, the computational problem rapidly increases with the size of the molecnle and the required accuracy of the expansion. 

This is a simple example of valence bond configuration interaction. 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many-Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamiltonian operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type, rather than the more general UHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of V . Various combinations of these assumptions result in final expressions that differ by factors of V2, V4, etc., from those given here. In the present chapter, the MOs are always spin-MOs, and conversion of a restricted summation to unrestricted is always noted explicitly. 

This is the oldest and perhaps the easiest method to understand, and is based on the variational principle (Appendix B), analogous to the FIF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (CI). The MOs used for budding the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D,T, etc., indicate determinants that are Singly, Doubly,Triply, etc., excited relative to the HF configuration. 

This is an example of a constrained optimization, the energy should be minimized under the constraint that the total Cl wave function is normalized. Introducing a Lagrange multiplier (Section 12.5), this can be written as 

The first bracket is the energy of the Cl wave function and the second bracket is the norm of the wave function. In terms of determinants (eq. (4.2)), these can be written as in eq. (4.4). 

The diagonal elements in the sum involving the Hamiltonian operator are energies of the corresponding determinants. The overlap elements between different determinants are zero as they are built from orthogonal MOs (eq. (3.20)). The variational procedure corresponds to setting all the derivatives of the Lagrange function (eq. (4.3)) with respect to the A expansion coefficients equal to zero. 

There are various ways to account for the electron correlation effects. In the following we will classify these methods into three groups configuration interaction, perturbation, and coupled cluster methods. 

In the Cl methods mentioned so far, only the mixing coefficients of the excited configurations are optimized in the variational calculations. If we optimize both the coefficients of the configurations and those of the basis functions, the method is called MCSCF, which stands for multiconfiguration self-consistent field calculation. One popular MCSCF technique is the complete active-space 

SCF (CASSCF) method, which divides all the molecular orbitals into three sets those doubly occupied orbitals which do not take part in the Cl calculations, those vacant orbitals which also do not participate in the Cl exercise, and those occupied orbitals and vacant orbitals that form the active space. The list of configurations that take part in the Cl calculation can be generated by considering all possible substitutions of the active electrons among the active orbitals. 

Finally, it should be stressed again that Cl calculations are variational i.e., the energy obtained cannot be lower than the exact energy, as stipulated by the variational principle. 

Here we are not concerned with the special reductions that apply to free atoms, and emphasize those general features of the theory that are equally applicable to systems of lower symmetry. We note that in molecular theory the term Cl calculation is often applied indiscriminately to describe the mixing of spin-orbital configurations within a given configuration or orbital configuration this is not, of course. Cl in the more precise sense of Condon and Shortley. 

It is clear that the expansion (3.1.4) will generally contain a very large number of terms, even when we admit s formed from a relatively small orbital basis. For example, if we consider a 5-electron system and use 10 basis orbitals then each spin-orbital configuration is defined by selecting 5 out of 20 spin-orbitals, and the full expansion will consequently contain 201/1515 or 15 564 terms. Even in this simple case, solution of the secular equations (3.1.8) requires a large computer and special techniques and the dimensions of the problem rise astronomically with further increase in the numbers of electrons and basis orbitals. 

Fortunately, however, it may turn out that many of the s my have symmetry properties different from that of the state being approximated, and according to a general theorem (A3.22) may be discarded and it may also turn out that many orbital configurations are of negligible 

First we consider the effect of symmetry, in particular of the spin symmetry that is present even for a molecule with no spatial symmetry. If we assume in first approximation a spin-free Hamiltonian, it is clear that H and the total spin operators 

Spatial symmetry may be utilized in a similar way. According to the theorems in Appendix 3 (p. 541), the expansion of any wavefunction W of given symmetry species contains only symmetry functions of the same species. The situation is precisely analogous to that which arises in the case of spin for the eigenvalues (S, M) are in fact the labels that define the different basis functions (Af = S, S - 1. -S) of a (2S + 1)-dimensional representation Dj of the group of rotations in spin space, and therefore correspond to the species labels (or, i) used in Appendix 3. Functions of pure symmetry species, with respect to spatial symmetry operations, may again be built up by linear combination of the basic determinants for molecules, this is easily accomplished by the methods of Appendix 3, and adequate illustrations appear in later sections. 

To overcome the deficiencies of the Hartree-Fock wave function (for example, improper behavior as internuclear distances go to infinity and very inaccurate dissociation energies), one can introduce configuration interaction (Cl), thus going beyond the Hartree-Fock approximation. Recall (Section 11.3) that in a molecular Cl calculation one begins with a set of basis functions Xi, does an SCF MO calculation to find SCF occupied and virtual (unoccupied) MOs, uses these MOs to form configuration (state) functions (CSFs) writes the molecular wave function i/r as a linear combination 2,- of the CSFs, and uses 

Each CSF is a linear combination of one to a few Slater determinants, is an eigenfunction of the spin operators and S, and satisfies the spatial symmetry requironents of the molecule. Alternatively, the Cl wave function can be expressed as the equivalent linear combination of Slater determinants. When this is done, the number of Slater determinants is typically 4 or 5 times the number of CSFs. 

The configuration functions in a Cl calculation are classified as singly excited, doubly excited, triply excited,. . ., according to whether 1, 2, 3,. . . electrons are excited from occupied to unoccupied (virtual) orbitals. For example, the H2 configuration function (T% s used in Eq. (13.95) is doubly excited (another term sometimes used is doubly substituted). 

The number of possible configuration functions with the proper symmetry increases extremely rapidly as the number of electrons and the number of basis functions increase. For n electrons and b basis functions, the number of configuration functions turns out to be roughly proportional to b . A Cl calculation that includes all possible configuration 

For a molecule with n electrons and with spin quantum number 5 = 0, the number of CSFs in a full Cl calculation (with spatial symmetry restrictions ignored) is Wilson, page 199) 

Having discussed ways to reduce the scope of the MCSCF problem, it is appropriate to consider the other limiting case. What if we carry out a CASSCF calculation for all electrons including all orbitals in the complete active space Such a calculation is called full configuration interaction or full CF. Witliin the choice of basis set, it is the best possible calculation that can be done, because it considers the contribution of every possible CSF. Thus, a full CI with an infinite basis set is an exact solution of the (non-relativistic, Bom-Oppenheimer, time-independent) Schrodinger equation. 

Note that no reoptimization of HF orbitals is required, since the set of all possible CSFs is complete . However, tliat is not much help in a computational efficiency sense, since the number of CSFs in a full CI can be staggeringly large. The trouble is not the number of electrons, which is a constant, but the number of basis functions. Returning to our methanol example above, if we were to use Hie 6-31G(d) basis set, the total number of basis functions would be 38. Using Eq. (7.9) to determine the number of CSFs in our (14,38) full CI we find that we must optunize 2.4 x 10 expansion coefficients ( ), and this is really a rather small basis set for chemical purposes. 

full CI calculations with large basis sets are usually carried out for only the smallest of molecules (it is partly as a result of such calculations that the relative contributions to basis-set quality of polarization functions vs. decontraction of valence functions, as discussed in Chapter 6, were discovered). In larger systems, the practical restriction to smaller basis sets makes full CI calculations less chemically interesting, but such calculations remain useful to the extent that, as an optimal limit, they permit an evaluation of the quality of other methodologies for including electron correlation using the same basis set. We turn now to a consideration of such other methods. 

If we consider all possible excited configurations that can be generated from the HF determinant, we have a full Cl, but such a calculation is typically too demanding to accomplish. However, just as we reduced the scope of CAS calculations by using RAS spaces, what if we were to reduce the Cl problem by allowing only a limited number of excitations How many should we include To proceed in evaluating this question, it is helpful to rewrite Eq. (7.1) using a more descriptive notation, i.e., 

If we assume that we do not have any problem with non-dynamical correlation, we may assume that there is little need to reoptimize the MOs even if we do not plan to carry out the expansion in Eq. (7.10) to its full Cl limit. In that case, the problem is reduced to determining the expansion coefficients for each excited CSF that is included. The energies E of N different Cl wave functions (i.e., corresponding to different variationally determined sets of coefficients) can be determined from the N roots of the Cl secular equation 

Given a set of one-electron functions tpi . (orbitals) which contains more functions than there are electrons, more than one determinant can be constructed. If the orbitals are physically meaningful, e.g., if they correspond to occupied hydrogen-like orbitals that can be called the electronic configuration of the system under study, we use the notion electronic configuration synonymously with Slater determinant. All possible Slater determinants form an N-particle basis set, into which the quantum mechanical state can be expanded. This expansion is called configuration interaction. 

Not all of these determinants are eigenfunctions of the total squared spin operator S, but one may construct linear combinations of determinants that are [287,288]. Then, the complexity of the problem can be reduced by definition of these so-called configuration state functions (CSF), 

The expansion coeeficients Cm for state A are called Cl coefficients. Hence, electronic ground and excited states A can be described on equal footing within a Cl-type approach by different sets of expansion parameters Ca = C/,4. Later in this chapter, we will see that these coefficients can be determined by diagonalization of the proper Hamiltonian matrix, which in turn yields the coefficient vectors Ca for as many electronic states as there are basis functions I j. 

In approximate Cl methods, the set of many-particle basis functions is restricted and not infinitely large, i.e., it is not complete. Then, the many-particle basis is usually constructed systematically from a given reference basis function. (such as the Slater determinant, which is constructed to approximate the ground state of a many-electron system in (Dirac-)Hartree-Fock theory). 

In this systematic construction process, spinors which enter the reference Slater determinant (these are the so-called occupied spinors) are substituted by new orthogonal spinors (which are, for instance, the virtual spinors that are obtained in a Dirac-Fock-Roothaan calculation see chapter 10). One then usually distinguishes sets of singly substituted many-electron functions 4 o, where an occupied spinor o, has been substituted by a virtual spinor from doubly substituted o]o j so on. The I o are also called single-excited determinants, the I oj-oy then double-excited determinants etc. It is important to understand that these excitations are not to be confused with the excited states of quantum mechanics they simply denote a substitution pattern. Eq. (8.100) can be rewritten in terms of the excitation hierarchy as 

There are several techniques for going beyond the SCF method and thereby including some effects of electron correlation. Some extremely accurate calculations on small atoms and molecules, making explicit use of interparticle coordinates, were described in Section 7-8. There is one general technique, however, that has traditionally been used for including effects of correlation in many-electron systems. This technique is called configuration interaction (Cl). 

The mathematical idea of Cl is quite obvious. Recall that we restricted our SCF wavefunction to be a single determinant for a closed-shell system. To go beyond the optimum (restricted Hartree-Fock) level, then, we allow the wavefunction to be a linear 

If we go through the mathematical formalism and express E as expand this as integrals over Di and D2, and require BEIdci = 0, we obtain the same sort of 2 X 2 determinantal equation that we find when minimizing an MO energy as a function of mixing of two AOs. That is, we obtain 

We see that, whereas before we might have had two AOs interacting to form two MOs, here we have two configurations (i.e., two determinantal functions) interacting to form two approximate wavefunctions. Our example involves only two configurations, but there is no limit to the number of configurations that can be mixed in this way. 

Since each configuration D contains products of MOs, each of which is typically a sum of AOs, the integrals Hij and Sij can result in very large numbers of integrals over basis functions when they are expanded. This is the sort of situation where a computer is essential, and Cl on atoms and molecules, while still expensive compared to SCF, have become routine on modem computers. 

FIGURE 3.2 Possible results of increasing the order of Moller-Plesset calculations. The circles show monotonic convergence. The squares show oscillating convergence. The triangles show a diverging series. 

There is also a local MP2 (LMP2) method. LMP2 calculations require less CPU time than MP2 calculations. LMP2 is also less susceptible to basis set superposition error. The price of these improvements is that about 98% of the MP2 energy correction is recovered by LMP2. 

A configuration interaction wave function is a multiple-determinant wave function. This is constructed by starting with the HF wave function and making new determinants by promoting electrons from the occupied to unoccupied or- 

The configuration interaction calculation with all possible excitations is called a full Cl. The full Cl calculation using an infinitely large basis set will give an exact quantum mechanical result. However, full Cl calculations are very rarely done due to the immense amount of computer power required. 

Cl results can vary a little bit from one software program to another for open-shell molecules. This is because of the HF reference state being used. Some programs, such as Gaussian, use a UHF reference state. Other programs, such as MOLPRO and MOLCAS, use a ROHF reference state. The difference in results is generally fairly small and becomes smaller with higher-order calculations. In the limit of a full Cl, there is no difference. 

C is a column matrix whose elements are the Cj in (6.68) and E are the eigenvalues. The other matrix elements are given by 

The question always arises as to how many configurations should be included. In the helium atom the radial correlation is improved by expanding in 5 orbitals, for example, 

The same formulation can also be used for the single determinant of the Hartree-Fock wavefunction. All coefEcients Sno then vanish and only the orbitals are unitarily transformed, i.e. 

The formulation of the MCSCF wavefunction in Eq. (9.47) will later be the starting point for the derivation of MCSCF linear response functions in Section 11.2. 

In the multiconfigurational self-consistent field method both the configuration coefficients C o as well as the molecular orbital coefEcients Cfip are varied until the energy becomes minimal. If one keeps the latter fixed and optimizes the energy only with respect to the configuration coefEcients C o, i - 

However, normally one expresses the Cl wavefunction like in Eq. (9.28) in terms of the Hartree-Fock wavefunction and the excited determinants . ) as 

Application of the variational condition, Eq. (9.54), then leads to a set of linear equations for the configuration coefEcients, which are conveniently written as the following matrix eigenvalue equation 

The Cl method is based on the expansion of the electronic wave function in a linear combination of Slater determinants P = 4 (X). including the ground-state Slater determinant and a number of substituted (or excited] Slater determinants  

Here is the ground-state Hartree-Fock wave function, the indexes i, j. denote occupied orbitals, the indexes a,b. denote virtual orbitals, and indicates the Slater determinant obtained from I o by replacing the occupied orbital i with the virtual orbital a. 

The ansatz wave function of Eq. (4.38) must solve the Schrbdinger equation 

The excitation energies can be found, within the Cl approach, by simply diagonalizing the Cl matrix in the left hand side of Eq. (4.40). Moreover, solutions of increasing accuracy can be systematically obtained by including higher excitation levels in the expansion in Eq. (4.38). When all possible substituted Salter determinant are considered (full Cl limit), the exact solution, within the given atomic-orbital space, is obtained. 

Despite its conceptual simplicity, the Cl method is often computationally unaffordable for most practical applications, because the dimension of the Cl matrix becomes rapidly huge, if high-order substitutions are considered for the Slater determinants in the expansion in Eq. (4.38). For this reason, calculations are generally performed at the Cl single (CIS) or Cl single and double (CISD) level of theory. In particular, despite its limited accuracy, the CIS approach is a rather common choice for the calculation of excited states because of its conceptual simplicity and computational efficiency. 

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Wlien first proposed, density llinctional theory was not widely accepted in the chemistry conununity. The theory is not rigorous in the sense that it is not clear how to improve the estimates for the ground-state energies. For wavefiinction-based methods, one can include more Slater detenuinants as in a configuration interaction approach. As the wavellmctions improve via the variational theorem, the energy is lowered. In density fiinctional theory, there is no... 

Bundgen P, Grein F and Thakkar A J 1995 Dipole and quadrupole moments of small molecules. An ab initio study using perturbatively corrected, multi-reference, configuration interaction wavefunctions J. Mol. Struct. (Theochem) 334 7... 

The magnitude of the perturbations can be calculated fairly quantitatively from high-quality electronic wavefunctions including configuration interaction [24]. 

Atomic natural orbital (ANO) basis sets [44] are fonned by contracting Gaussian fiinctions so as to reproduce the natural orbitals obtained from correlated (usually using a configuration interaction with... 

First-principles models of solid surfaces and adsorption and reaction of atoms and molecules on those surfaces range from ab initio quantum chemistry (HF configuration interaction (Cl), perturbation theory (PT), etc for details see chapter B3.1 ) on small, finite clusters of atoms to HF or DFT on two-dimensionally infinite slabs. In between these... 

NakatsujI H and Nakal H 1990 Theoretical study on molecular and dissociative chemisorptions of an O2 molecule on an Ag surface dipped adcluster model combined with symmetry-adapted cluster-configuration interaction method Chem. Phys. Lett. 174 283-6... 

Werner H-J 1987 Matrix-formulated direct multiconfigurational self-consistent field and multi reference configuration interaction methods Adv. Chem. Phys. 69 1... 

Eq. (15b) for OH + H2 using multi reference configuration interaction wave functions. 

Yon can use a sin gle poin t calculation that determines energies for ground and excited states, using configuration interaction, to predict frequencies and intensities of an electron ic ultraviolet-visible spectrum. 

HyperChem always com putes the electron ic properties for the molecule as the last step of a geometry optimization or molecular dyn am ics calcu lation. However, if you would like to perform a configuration interaction calculation at the optimized geometry, an additional sin gle poin t calcu lation is requ ired with theCI option being turned on. 

Some of the ways in which excitai-state wavefunctions can be included in a configuration interaction Illation (Figure adapted from Hehre W ], L Roikiin, P zi R Schleyer and ] A Hehre 1986. Ab initio Molecular aital Theory. New York, Wiley.)... 

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