Configuration state function

As discussed in Section 2.4, the exact nonrelativistic wave function is an eigenfunction of the total and projected spins but Slater determinants are eigenfunctions of the projected spin only. Since it is often advantageous to employ a basis of spin eigenfunctions in approximate calculations, we shall in this section see how we can set up a basis of spin eigenfunctions by taking linear combinations of Slater determinants. 

The lack of spin symmetry in the determinants arises since the spin-orbital ON operators - of which the determinants are eigenfunctions - do not commute with the operator for the total spin see (2.4.9) and (2.4.10). By contrast, the orbital ON operators 

We can therefore set up a basis of functions that are simultaneously eigenfunctions of the orbital ON operators as well as the operators for the projected and total spins. Such spin-adapted functions are called configuration state fimctions (CSFs) [2], 

Let us investigate the eigenfunctions of the m-bital ON qrerators in greater detail. We first note that Slater determinants are eigenfunctions of the orbital ON operators  

we note that different determinants may have the same orbital occupation numbers but different spin-orbital occupation numbers since an orbital occupation equal to 1 may represent either the occupation of an alpha spin orbital or the occupation of a beta spin orbital by an unpaired electron. The set of all determinants with the same orbital occupation numbers but different spin-orbital occupation numbers is said to constitute an orbital configuration. In a sense. 

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

Note that this is also a functional of liaAr), Cas(r), and 4 ). Imposing constraints concerning the orthonormality of the configuration state function (C) and one-particle orbitals (pi) on the equation, one can derive the Eock operator from. A based on the variational principle ... 

Figure 4.3 Forming configurational state functions from Slater determinants...

With this choice for H°, equations (7) and (8) are automatically valid for the perturbation. The only restriction is that we have to use orthogonal orbitals and Slater determinants rather than Configuration State Functions (CSFs) as a basis for the perturbation. None of these restrictions is constraining, however. 

Figure 2. The P,T-odd interaction constant versus the number of configuration state functions (CSFs) for the YbF molecule.

The basis of the expansion, ifra, are configuration state functions (CSF), which are linear combinations of Slater determinants that are eigenfunctions of the spin operator and have the correct spatial symmetry and total spin of the electronic state under investigation [42],... 

T is expanded in a basis of time-reversal adapted configuration state functions [8] (TRA-CSFs, x F)... 

The selection of configuration state functions to be included in MCSCF calculations is not a trivial task. Two approaches which can reduce the complexity of the problem are the complete active space self-consistent-field (CASSCF) [68] and the restricted active space self-consistent-field (RASSCF) [69] approach. Both are implemented in the Dalton program package [57] and are used in this study. Throughout the paper a CASSCF calculation is denoted by i active gactive RASSCF calculation by For the active spaces of HF, H2O, and CH4... 

The CASSCF procedure (a special case of MC methods) is a full Cl calculation among all possible configuration state functions (CSFs), which arise from the distribution of a certain number of electrons (active electrons) in a certain... 

Let us assume that the orbit-generating wavefunction for the nth state is given in the form of a configuration interaction wavefunction (i.e., a linear combination of configuration state functions) ... 

It has been demonstrated that a given electronic configuration can yield several space- and spin- adapted determinental wavefunctions such functions are referred to as configuration state functions (CSFs). These CSF wavefunctions are not the exact eigenfunctions of the many-electron Hamiltonian, H they are simply functions which possess the space, spin, and permutational symmetry of the exact eigenstates. As such, they comprise an acceptable set of functions to use in, for example, a linear variational treatment of the true states. 

The resultant family of six electronic states can be described in terms of the six configuration state functions (CSFs) that arise when one occupies the pair of bonding o and antibonding o molecular orbitals with two electrons. The CSFs are combinations of Slater determinants formed to generate proper spin- and spatial symmetry- functions. 

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