CSFs

For these reasons, in the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g. a few to several thousand) chosen to describe essential correlations (i.e. configuration crossings, near degeneracies, proper dissociation, etc, all of which are often tenned non-dynamicaI correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc nature that are important when low-lying virtual orbitals are present). 

In this approach [ ], the LCAO-MO coefficients are detemiined first via a smgle-configuration SCF calculation or an MCSCF calculation using a small number of CSFs. The Cj coefficients are subsequently detemiined by making the expectation value ( P // T ) / ( FIT ) stationary. 

The Cl wavefiinction is most connnonly constmcted from CSFs of (bj that include ... 

The orbitals from which electrons are removed can be restricted to focus attention on the correlations among certain orbitals. For example, if the excitations from the core electrons are excluded, one computes the total energy that contains no core correlation energy. The number of CSFs included in the Cl calculation can be far in excess of the number considered in typical MCSCF calculations. Cl wavefimctions including 5000 to 50 000 CSFs are routine, and fimctions with one to several billion CSFs are within the realm of practicality [53]. 

The need for such large CSF expansions should not be surprising considering (i) that each electron pair requires at least two CSFs to fomi polarized orbital pairs, (ii) there are of the order of N N - 1 )/2 = X electron pairs for N electrons, hence (iii) the number of temis in the Cl wavefiinction scales as 1. For a molecule containing ten electrons, there could be 2 =3.5x10 temis in the Cl expansion. This may be an overestimate of the number of CSFs needed, but it demonstrates how rapidly the number of CSFs can grow with the number of electrons. 

In the MPPT/MBPT method, once the reference CSF is chosen and the SCF orbitals belonging to this CSF are detennined, the wavefiinction T and energy E are detennined in an order-by-order maimer. The perturbation equations determine what CSFs to include and their particular order. This is one of the primary strengdis of this technique it does not require one to make fiirtlier choices, in contrast to the MCSCF and Cl treatments where one needs to choose which CSFs to include. 

These quartic equations are solved in an iterative maimer and, as such, are susceptible to convergence difficulties. In any such iterative process, it is important to start with an approximation reasonably close to the final result. In CC theory, this is often achieved by neglecting all of tlie temis tliat are nonlinear in the t amplitudes (because the ts are assumed to be less than unity in magnitude) and ignoring factors that couple different doubly-excited CSFs (i.e. the sum over i, f, m and n ). This gives t amplitudes that are equal to the... 

These approaches provide alternatives to the conventional tools of quantum chemistry. The Cl, MCSCF, MPPT/MBPT, and CC methods move beyond the single-configuration picture by adding to the wavefimction more configurations whose amplitudes they each detennine in their own way. This can lead to a very large number of CSFs in the correlated wavefimction and, as a result, a need for extraordinary computer resources. 

So-called complete active space (CAS) methods fomi all CSFs that... 

Methods that are based on making the fiinctional (T // T ) / ( T T ) stationary yield upper bounds to the lowest energy state having the synnnetry of the CSFs in T. The Cl and MCSCF methods are of this type. 

By carefully adjustmg the variational wavefiinction used, it is possible to circumvent size-extensivity problems for selected species. For example, the Cl calculation on Bc2 using all 2 CSFs fomied by placing the four valence electrons into the 2a, 2a, 30g, 3a, In, and iTt orbitals can yield an energy equal to twice that of the Be atom described by CSFs in which the two valence electrons of the Be atom are placed into the... 

D) MOST PERTURBATION AND CC METHODS ARE SIZE-EXTENSIVE, BUT DO NOT PROVIDE UPPER BOUNDS AND THEY ASSUME THAT ONE CSF DOMINATES... 

This can be seen by considering the second-order MPPT energy of two non-interacting Be atoms. The reference CSF is = lsi 2s- ls 2s as discussed earlier, only doubly-excited CSFs contribute to the... 

As larger atomic basis sets are employed, the size of the CSF list used to treat a dynamic correlation increases rapidly. For example, many of the above methods use singly- and doubly-excited CSFs for this purpose. For large basis sets, the number of such CSFs (N ) scales as the number of electrons squared uptimes the number... 

MCSCF methods describe a wave function by the linear combination of M configuration state functions (CSFs), with Cl coefficients, Ck,... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

A simple example would be in a study of a diatomic molecule that in a Hartree-Fock calculation has a bonded cr orbital as the highest occupied MO (HOMO) and a a lowest unoccupied MO (LUMO). A CASSCF calculation would then use the two a electrons and set up four CSFs with single and double excitations from the HOMO into the a orbital. This allows the bond dissociation to be described correctly, with different amounts of the neutral atoms, ion pair, and bonded pair controlled by the Cl coefficients, with the optimal shapes of the orbitals also being found. For more complicated systems... 

I. CSFs Are Used to Express the Full N-Eleetron Wavefunetion... 

It has been demonstrated that a given eleetronie eonfiguration ean yield several spaee- and spin- adapted determinental wavefunetions sueh funetions are referred to as eonfiguration state funetions (CSFs). These CSF wavefunetions are not the exaet eigenfunetions of the many-eleetron Hamiltonian, H they are simply funetions whieh possess the spaee, spin, and permutational symmetry of the exaet eigenstates. As sueh, they eomprise an aeeeptable set of funetions to use in, for example, a linear variational treatment of the true states. 

In sueh variational treatments of eleetronie strueture, the N-eleetron wavefunetion F is expanded as a sum over all CSFs that possess the desired spatial and spin symmetry ... 

Here, the d>j represent the CSFs that are of the eorreet symmetry, and the Cj are their expansion eoeffieients to be determined in the variational ealeulation. If the spin-orbitals used to form the determinants, that in turn form the CSFs Oj, are orthonormal one-eleetron funetions (i.e., <(l)k (l)j> = 5kj), then the CSFs ean be shown to be orthonormal funetions of N eleetrons... 

The above expansion of the full N-eleetron wavefunetion is termed a "eonfiguration-interaetion" (Cl) expansion. It is, in prineiple, a mathematieally rigorous approaeh to expressing F beeause the set of aU determinants that ean be formed from a eomplete set of spin-orbitals ean be shown to be eomplete. In praetiee, one is limited to the number of orbitals that ean be used and in the number of CSFs that ean be ineluded in the Cl expansion. Nevertheless, the Cl expansion method forms the basis of the most eommonly used teehniques in quantum ehemistry. 

Within a variational treatment, the relative eontributions of the spin-and spaee-symmetry adapted CSFs are determined by solving a seeular problem for the eigenvalues (Ei) and eigenveetors (C ) of the matrix representation H of the full many-eleetron Hamiltonian H within this CSF basis ... 

The eigenvalue Ei gives the variational estimate for the energy of the i state, and the entries in the eorresponding eigenveetor Ci k give the eontribution of the CSF to the wavefunetion Fi in the sense that... 

II. The Slater-Condon Rules Give Expressions for the Operator Matrix Elements Among the CSFs... 

Beeause the CSFs are simple linear eombinations of determinants with eoeffieients determined by spaee and spin symmetry, the Hi j matrix in terms of determinants ean be used to generate the Hk,l matrix over CSFs. 

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