Singly excited CSF

That singly excited CSFs allow for orbital relaxation can be seen as follows. Consider a wavefunction consisting of one CSF l(f>i... (f>j... (f l to which singly excited CSFs of the form l( ) ]... (f>m... ()xl have been added with coefficients Qjm ... 

The sum of CSFs that are singly excited in the ith spin-orbital with respect to. .. < 1 is therefore seen to allow the spin-orbital (f>i to relax into the new spin-orbital It is in this sense that singly excited CSFs allow for orbital reoptimization. 

In summary, doubly excited CSFs are often employed to permit polarized orbital pair formation and hence to allow for electron correlations. Singly excited CSFs are included to permit orbital relaxation (i.e., orbital reoptimization) to occur. 

A somewhat special case is the matrix element between the HF determinant and a singly excited CSF. The Condon-Slater rules applied to this situation dictate that... 

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... 

When the states P1 and P2 are described as linear combinations of CSFs as introduced earlier ( Fi = Zk CiKK), these matrix elements can be expressed in terms of CSF-based matrix elements < K I eri IOl >. The fact that the electric dipole operator is a one-electron operator, in combination with the SC rules, guarantees that only states for which the dominant determinants differ by at most a single spin-orbital (i.e., those which are "singly excited") can be connected via electric dipole transitions through first order (i.e., in a one-photon transition to which the < Fi Ii eri F2 > matrix elements pertain). It is for this reason that light with energy adequate to ionize or excite deep core electrons in atoms or molecules usually causes such ionization or excitation rather than double ionization or excitation of valence-level electrons the latter are two-electron events. 

The use of doubly excited CSFs is thus seen as a mechanism by which VF can place electron pairs, which in the single-configuration picture occupy the same orbital, into... 

The second step of the calculation involves the treatment of dynamic correlation effects, which can be approached by many-body perturbation theory (62) or configuration interaction (63). Multireference coupled-cluster techniques have been developed (64—66) but they are computationally far more demanding and still not established as standard methods. At this point, we will only focus on configuration interaction approaches. What is done in these approaches is to regard the entire zeroth-order wavefunc-tion Tj) or its constituent parts double excitations relative to these reference functions. This produces a set of excited CSFs ( Q) that are used as expansion space for the configuration interaction (Cl) procedure. The resulting wavefunction may be written as... 

H is the Hamiltonian operator and the numbering of the CSFs is arbitrary, but for convenience we will take I l = I hf and then all singly excited determinants, all doubly excited, etc. Solving the secular equation is equivalent to diagonalizing H, and permits determination of the CI coefficients associated with each energy. While this is presented without derivation, the formalism is entirely analogous to that used to develop Eq. (4.21). 

This is called the first-order variational space which is spanned by the reference state, the orthogonal complement states and the single excitation states (10>, ln>, T 10> or equivalently by the set of expansion CSFs and the single excitation states n>,r 0>. The parameters K and p may be determined by minimizing the expectation value of the Hamiltonian operator within this non-orthonormal basis. This results in the non-orthogonal matrix eigenvalue equation... 

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