INDEX Slater determinants

The anti symmetrized orbital produet A (l)i(l)2Cl)3 is represented by the short hand (1>1(1>2(1>3 I and is referred to as a Slater determinant. The origin of this notation ean be made elear by noting that (1/VN ) times the determinant of a matrix whose rows are labeled by the index i of the spin-orbital (jii and whose eolumns are labeled by the index j of the eleetron at rj is equal to the above funetion A (l)i(l)2Cl)3 = (1/V3 ) det(( )i (rj)). The general strueture of sueh Slater determinants is illustrated below ... 

The term D(j) can be taken as a Slater determinant, formed by n functions chosen from a set of in available spinorbitals, and ordered following the actual internal values of the j index vectors. That is ... 

Inequation (18) the D(j k) terms are n-electron Slater determinants formed by the spin-orbitals numbered by means of the direct sum j0k of the vector index parameters attached to the involved nested sums and to the occupied-unoccupied orbitals respectively. That is ... 

The components of vectors D k.=i..m are completely defined by the parameters of the underlying full-CI type wave function, and the index sets of Slater-determinants and their subdeterminants according to (13). The number of vectors D is ( ), and this is of course equal to the number of geminals g constructed over the M-dimensional one-particle basis Bm-... 

The reference state T is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions orbital energy functional E = To + Ec is to be made stationary, subject to the orbital orthonormality constraint (i j) = StJ, imposed by introducing a matrix of Lagrange multipliers. The variational condition is... 

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. 

Hartree-Fock Scheme.- For the sake of simplicity, we will consider the eigenvlue relations (2.3) for a non-degenerate eigenvalue X, for which 0. If Ca and Da are two approximate solutions - characterized by the index "a"- the best approximations are obtained by using the bi-variational principle (B.3.6). In the Hartree-Fock scheme for N fermions, the approximate solutions are represented by two Slater determinants ... 

In this expression the subscript 0 denotes the zeroth-order function d>A >B where < > and d>B are SCF wavefunctions (single Slater determinants in the closed-shell case) of the respective monomers . The index p runs over all single and double excitations from the zero-order function. The same expression is also used by Magnasco and Musso ° but in this case index p runs over singly excited configurations only dispersion effects are not included in their treatment. In addition the unoccupied orbitals are expressed in the form of antibonding orbitals. 

For a free electron gas, it is possible to evaluate the Hartree-Fock exchange energy directly [3,16], The Slater determinant is constmcted using free electron orbitals. Each orbital is labelled by a k and a spin index. The Coulomb... 

On the right a classification of the contributing entries is given which refers to the character of the single particle indices. We classify an index r as a particle index (p) if it refers to a single particle orbital ipr x) which is vacant (virtual) in the Slater determinant ). Conversely a hole index (ft) labels an occupied orbital in ). Accordingly the first component of in... 

Here, rir denotes the occupation number of the orbital with index r in the ground state Slater determinant J ), while = 1 —is the anti-occupation number. The eigenvalues of this hermitian matrix yield approximations for the excitation energies of the system under study while the eigenvectors can be used to approximate transition moments. Since the FOSEP approximation has been studied in detail in reference [21], we will only summarise the most important properties in this paragraph. 

The second order similarity index is generally defined by eq. (147) straightforwardly arising from the definition of the first order index r g by replacing the first order density matrices p (1), pg(1) by the second order or pair density matrices p (1,2), pg(1,2). Within the framework of the model where the wave functions and Og are approximated by the Slater determinants (134), the pair... 

Quantitative measures of the compactness of an A-electron wave function have been reported in Ref. [18] by means of the informational content Qc (or Shannon entropy) within the traditional Cl expansion method based on Slater determinants classified according to the excitation level with respect to a given reference determinant. Assuming that the A-electron wave function is normalized to unity Ea(S2) ICa(S2)P = 1), the counterpart formulation of that index for the seniority-based Cl approach is... 

The values of this index quantify the multiconfigura-tional character of the Cl wave function but do not report any detailed information on the contributions corresponding to different seniority subspaces. For Cl ( 2) expansions involving several values of the S2 index, we can define a weight Wq, which groups the contributions of all the Slater determinants with given seifiority number 2 in that expansion... 

One can also consider the distribution of each S2 subspace in terms of its corresponding Slater determinants and calculate its specific entropic index, which can be evaluated by means of the relationship... 

The results reported in Table 1 also allow one to compare, in terms of the values of the indices ( and lyy, the expansions of the wave fnnctions of these systems according to the molecular orbital basis sets in which they are expressed. As can be seen from that table, the values of both indices are considerably lower in the NO and basis sets than in their CMO counterparts (except for the Be atom in the STO-3G basis set) the Be atom recovers the improvement in the and NO molecular basis sets when the larger cc-pVDZ basis set is used. These results again confirm that the NO and molecular basis sets lead to more compact wave functions, as has been reported in Refs. [6, 9, 11]. These valnes also point out that the Ic and % indices constitute suitable devices to describe quantitatively the compactness of a wave function. The high values found for the indices in the Be and Mg atoms in the three molecular basis sets can be interpreted in terms of the strong correlation exhibited by those systems. The appropriate ground-state wave functions for these atoms reqnire several dominant Slater determinants. The values reported in Table 2 reflect that seniority levels with very low contribution to the wave functions can present a broad determinantal distribution, i.e., the Li2 molecule exhibits 7 2=4 > 5 values because its W =4 = 10 " weight is expanded on 7560 Slater determinants in the STO-3G basis set [6]. Moreover, the Ia=o index values reported in that table indicate that aU systems possess a narrower... 

Here is the ground-state Hartree-Fock wave function, the indexes i, j,... denote occupied orbitals, the indexes a,b,... denote virtual orbitals, and indicates the Slater determinant obtained from I o by replacing the occupied orbital i with the virtual orbital a. 

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... 

The second approach relies on the expansion of the SC wave function in Slater determinants constructed from non-orthogonal orbitals. The density matrix is calculated from cofactors, constructed in situ using graphical indexing techniques. Calculations have been carried out with this strategy for up to 14 active electrons. It has also been extended recently to include simultaneous optimization of the inactive electrons. ... 

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