Slater determinants spin projection

This difficulty is overcome with the aid of a projection operator by projecting out from the Slater determinant the component with the desired multiplicity 25+1, annihilating all other contaminating components. This can be done either after an already performed calculation (spin projection after variation, UHF with annihilation), or, as Lowdin has pointed out, one would expect a more negative total energy if the variation is performed with an already spin-projected Slater determinant [spin projection before variation, spin-projected extended Hartree-Fock (EHF) method]. The reason is that a spin-projected Slater determinant is a given linear combination of different Slater determinants. The variation in the expectation value of the Hamiltonian formed with a spin-projected Sater determinant thus provides equations (EHF equations), whose solutions represent the solution of this particular multiconfigura-tional SCF problem. 

Slater determinants with projected spin are, 1,1), , 0, and —j, 0, j). Our task is to expand the three genealogical CSFs in these three determinants. 

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 the ordinary Hartree-Fock scheme, the total wave function is approximated by a single Slater determinant and, if the system possesses certain symmetry properties, they may impose rather severe restrictions on the occupied spin orbitals see, e.g., Eq. 11.61. These restrictions may be removed and the total energy correspondingly decreased, if instead we approximate the total wave function by means of the first term in the symmetry adapted set, i.e., by the projection of a single determinant. Since in both cases,... 

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

This projection/annihilation approach is probably more useful as an analytical tool, for annihilating the principal spin contaminants from a wave function by hand calculation, for example, than as a computational tool. There is a vast body of literature (see, for example, Pauncz [18]) on generating spin eigenfunctions as linear combinations of Slater determinants, from explicitly precomputed Sanibel coefficients to diagonalizing the matrix of S. However, there are other methods that exploit the group theoretical structure of the problem more effectively, and we shall now turn to these. 

Some of the most efficient algorithms currently available for spin-coupled calculations involve the expansion of the SC wavefunction in terms of Slater determinants, leading to a summation over (N-p) -dimensional cofactors for the p-particle density matrix [23]. For calculations involving the full spin space, a further saving can be achieved by the use of projected spin functions [24], An alternative strategy is provided by CASVB (see Sect. 5). 

The UHF ansatz is necessary because in case of neutral solitons one has to deal with a doublet state. Thus a DODS (different orbitals for different spins) ansatz, as the UHF one, is necessary to describe the system. However, in the UHF method described so far, one Slater determinant with different spatial orbitals for electrons of different spins is applied, which is not an eigenfunction of S2, i.e. S(S+l)h2. The best way to overcome this difficulty would be to use the PHF (Projected Hartree Fock) method, also called EHF method (Extended Hartree Fock) where before the variation the correct spin eigenfunction is projected out of the DODS ansatz Slater determinant [66,67a]. Unfortunately numerical solution of the rather complicated EHF equations in each time step seems to be too tedious at present. Moreover for large systems the EHF wavefunction approaches the UHF one [68], however, this might be due to the approximations used in [67a]. Another possibility is to apply the projection after the variation using again Lowdin s projection operator [66]. Projection and annihilation techniques were... 

Here a given function involves an n-fold product of MOs, to which is applied some projection operator or operators 0. As electrons are fermions, the solutions to Eq. (2) will be antisymmetric to particle interchange, and it is usually convenient to incorporate this into the n-particle basis, in which case the will be Slater determinants. The Hamiltonian given in Eq. (1) is also spin-independent and commutes with all operations in the molecular point group, so that projection operators for particular spin and spatial symmetries could also appear in 0. The O obtained in this way are generally referred to as configuration state functions (CSF s). 

To overcome some of the problems inherent in the UHF method, it is possible to derive SCF equations based on minimizing the energy of a wavefimction formed by spin projecting a single Slater determinant starting... 

When an DODS Slater determinant of n electron pairs is projected on a spin space of spin quantum number S, its projection takes the form of a linear... 

As was mentioned in Sect. 4.1.3 in the unrestricted Hartree ock approximation (where the coordinate dependence of spin-up and spin-down MOs was allowed to differ) the one-determinant many-electron wavefunction is, in the general case, not an eigenfunction of the total spin operator S. To repair that deficiency the technique of projection is used [103] so that the resulting wavefunction becomes a sum of several Slater determinants and therefore partly takes into account electron-correlation, i.e. goes beyond the one-derminant HF approximation. However, the coefficients in the sum of Slater determinants are defined only by the projection procedure, i.e. the total spin-symmetry requirements introduced for the many-electron wavefunction. 

Closing both sides of Eq. (1) by an N-electron Slater determinant A of spin projection quantum number, one obtains... 

Different schemes have been devised to remove contaminants from higher spin states from UHF wavefunctions by means of the spin projection operators, which were introduced originally by Lbwdin. Removing high-energy spin states does lower the energies of UHF wavefunctions. However, these procedures lead to projected UHF (PUHF) wavefunctions that consist of several Slater determinants whose coefficients (cf. Eq, [4]) in been optimized variationally. 

If any two spin-orbitals are the same the projected function simply vanishes. This vanishing is the basis of what is usually called the Pauli exclusion principle. The function (O Eq. 2.131) is clearly a determinant of spin-orbitals with the spin-orbital index designating a row (column) and the electron numbering designating a column (row). This was first recognized by Slater and so such determinants are called Slater determinants and often denoted by the shorthand... 

Such Slater determinants are not themselves always spin eigenfunctions but they are eigenfunctions of Sz. They can be either coupled or projected to yield spin eigenfunctions for a particular Ms value and the resulting functions can then be further projected by the step-up... 

In the special case where the spin-orbitals are orthonormal and the trial functions are Slater determinants the expressions for the projective reduction coefficients are both simple and limited, given by Slaters rules to be discussed in detail in later chapters in this work. With such a choice there are Hamiltonian matrix elements between functions that differ from each other only in two or fewer orbitals and Mu = 6u- The expressions for these coefficients when the orbitals are not orthogonal involve the overlap integrals S, j between all the orbitals in the functions and there is no limitation on orbital differences between the functions and Mu is not the unit matrix. Every electronic permutation must be considered in their evaluation. For non-orthogonal orbitals it is thus much more difficult to consider systems with more than a few electrons and, because atomic orbitals on different centers are not orthogonal, this difficulty has hindered the development of VB theory in a quantitative manner until very recently. An account of modern VB developments forms a later part of this handbook. Usually LCAO MOs are developed so as to be orthogonal so that given... 

Table 11.1 The number of Slater determinants A det with spin projection zero obtained by distributing 2k electrons among 2k orbitals...

This relation follows from the observation that the spin-projection operator (2.2.30) is a linear combination of spin-orbital ON operators. Thus, since the ON operators commute among themselves (1.3.4), they must also commute with the spin-projection operator. From the commutation relations (2.4.5) and from the observation that there are no degeneracies among the spin-orbital occupation numbers (2.4.4), we conclude that the Slater determinants are eigenfunctions of the projected spin ... 

Although Slater determinants are not by themselves spin eigenfunctions, it is possible to determine spin eigenfunctions as simple linear combinations of determinants. A clue to the procedure for generating spin-adapted determinants is obtained from the observation that both the total- and projected-spin operators commute with the sum of the ON operators for alpha and beta spins ... 

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. 

We also need a systematic way to represent the Slater determinants belonging to a given configuration. Following the scheme for CSFs, we represent each determinant by a vector P, where the element Pj gives the total spin projection of the first i spin orbitals. Alternatively, the determinant may be represented by a vector p, where the element p,- is equal to the spin projection of spin orbital i (i.e. —5 or -I-5). Three determinants arise when-three spin orbitals are combined to give... 

Once our N-electron CSF has been reduced to a linear combination of one or two (N — 1)-electron spin eigenfunctions each multiplied by a creation operator, we may go one step further and expand each (N — l)-electron state in two (N — 2)-electron spin eigenfunctions as dictated by the penultimate element t -i in the genealogical coupling vector t. After N — I such steps, we arrive at an expansion in terms of determinants with projected spin M. In this way, we are led directly to an expansion of CSFs in Slater determinants where the coefficients are products of the genealogical coupling coefficients in (2.6.5) and (2.6.6). 

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