Slater determinant doubly

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). 

The first-order MPPT wavefunction can be evaluated in terms of Slater determinants that are excited relative to the SCF reference function k. Realizing again that the perturbation coupling matrix elements I>k H i> are non-zero only for doubly excited CSF s, and denoting such doubly excited i by a,b m,n the first-order... 

In Chapter 6, I discussed the open-shell HF-LCAO model. 1 considered the simple case where we had ti doubly occupied orbitals and 2 orbitals all singly occupied by parallel spin electrons. The ground-state wavefunction was a single Slater determinant. I explained that it was possible to derive an expression for the electronic energy... 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

How are the additional determinants beyond the HF constructed With N electrons and M basis functions, solution of the Roothaan-Hall equations for the RHF case will yield N/2 occupied MOs and M — N/2 unoccupied (virtual) MOs. Except for a minimum basis, there will always be more virtual than occupied MOs. A Slater detemfinant is determined by N/2 spatial MOs multiplied by two spin functions to yield N spinorbitals. By replacing MOs which are occupied in the HF determinant by MOs which are unoccupied, a whole series of determinants may be generated. These can be denoted according to how many occupied HF MOs have been replaced by unoccupied MOs, i.e. Slater determinants which are singly, doubly, triply, quadruply etc. excited relative to the HF determinant, up to a maximum of N excited electrons. These... 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF 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 (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

The matrix elements between the HF and a doubly excited state are given by two-electron integrals over MOs (eq. (4.7)). The difference in total energy between two Slater determinants becomes a difference in MO energies (essentially Koopmans theorem), and the explicit formula for the second-order Mpller-Plesset correction is... 

A single Slater determinant with N+ N represents a pure spin state if, and only if, the number of doubly filled orbitals defined by Eq. 11.57 equals AL. 

This case is, of course, realized if N orbitals are doubly occupied and the remaining (N+ N ) orbitals are all occupied by electrons having plus spin. By using Eq. 11.58, we can very easily check that a Slater determinant constructed in this way is actually an eigenfunction to S2 associated with the quantum number S — i(N+-N ). 

More abstractly the condition Tr (p+p ) = iV implies that the part of the Hilbert space defined by the projection operator p should be fully contained in the part defined by the projection operator p+. If we now vary p slightly so that this condition is no longer fulfilled, Eq. II.GO shows that the pure spin state previously described by the Slater determinant becomes mixed up with states of higher quantum numbers S = m+1,. . . . The idea of the electron pairing in doubly occupied orbitals is therefore essential in the Hartree-Fock scheme in order to secure that the Slater determinant really represents a pure spin state. This means, however, that, in the calculation of the best spin orbitals y>k(x), there is a new auxiliary condition of the form... 

In Eq. (1.16a), A is the antisymmetrizer operator that converts the orbital product into a Slater determinant, insuring satisfaction of the Pauli exclusion principle. In this equation (alone), the same spatial orbital might appear twice, with different indices to indicate the change in spin. For example, / i (0,(7 ypf HA) might be the same as i<0)(F K/>,0,0" 2). a doubly occupied spatial orbital (n]m> = 2), with a bar denoting opposite spin in the second spin-orbital. 

Let us now consider a simple example four noninteracting fermions in a onedimensional box, x e [0, 7r], The wave function is a Slater determinant with two doubly occupied orbitals ... 

In the absence of the interaction U, the ground state is that of uncorrelated electrons j o) and has the form of a Slater determinant. As U is turned on, the weight of doubly occupied sites must be reduced because they cost an additional energy U per site. Accordingly, the trial Gutzwiller wave function (GWF) I/g) is built from the Hartree-like uncorraleted wave function (HWF) l/o),... 

In open-shell electronic states, the orbitals are not all doubly occupied, and the preceding procedure is not applicable. However, if the wave function can be written as a single Slater determinant, one can use a modified procedure to obtain energy-localized MOs here also. The procedure is to deal with the a spin-orbitals and the jS spin-orbitals separately, using two different unitary transformation matrices Ba and B in (2.85). 

Instead, practical methods involve a subset of possible Slater determinants, especially those in which two electrons are moved from the orbitals they occupy in the HF wavefunction into empty orbitals. These doubly excited determinants provide a description of the physical effect missing in HF theory, correlation between the motions of different electrons. Single and triple excitations are also included in some correlated ab initio methods. Different methods use different techniques to decide which determinants to include, and all these methods are computationally more expensive than HF theory, in some cases considerably more. Single-reference correlated methods start from the HF wavefunction and include various excited determinants. Important methods in inorganic chemistry include Mpller-Plesset perturbation theory (MP2), coupled cluster theory with single and double excitations (CCSD), and a modified form of CCSD that also accounts approximately for triple excitations, CCSD(T). 

As we know from Section III, the Slater determinant for a doubly excited configuration can be written as... 

To illustrate the modifications of UHF formalism, it is convenient to consider pure spin symmetry for a single Slater determinant with Nc doubly occupied spatial orbitals Xi and N0 singly occupied orbitals y". The corresponding UHF state has Na mj = occupied spin orbitals and Np rns = — J, occupied spin orbitals f. The number of open-shell and closed-shell orbitals are, respectively Na = Na — Np > 0 and Nc = Np. Occupation numbers for the spatial orbitals are nc = 2, n ° = 1. If all orbital functions are normalized, a canonical form of the RHF reference state is defined by orthogonalizing the closed- and open-shell sets separately. 

In Equation 6.17, each orbital listed in the Slater determinant contains two electrons of opposite spin (in accordance with the Pauli principle), and the occupancies with [3 spin electrons are indicated by overlinings. 0 thus contains nt2 doubly occupied spatial orbitals. 

Since spin-orbit coupling is very important in heavy element compounds and the structure of the full microscopic Hamiltonians is rather complicated, several attempts have been made to develop approximate one-electron spin-orbit Hamiltonians. The application of an (effective) one-electron spin-orbit Hamiltonian has several computational advantages in spin-orbit Cl or perturbation calculations (1) all integrals may be kept in central memory, (2) there is no need for a summation over common indices in singly excited Slater determinants, and (3) matrix elements coupling doubly excited configurations do not occur. In many approximate schemes, even the tedious four-index transformation of two-electron integrals ceases to apply. The central question that comes up in this context deals with the accuracy of such an approximation, of course. 

Firstly, the function (70) is invariant under a linear transformation of the m doubly occupied orbitals amongst themselves. A proof of this statement seems hardly necessary as, in the case m = N, equation (70) is equivalent to a Slater determinant, and this property of a determinant is well-known. The m orbitals m may therefore be orthogonalized amongst themselves by a linear transformation, without altering the total wavefunction. This, of course, may be done in several ways, by transforming to MOs for example, but perhaps the most convenient method is to employ the Lowdin symmetric orthogonalization method 73... 

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