Spin-orbitals indices

PatxiR) 9- ( )> defined through active spin-orbital indices as / X (r)> to calculate the triply excited contribution Eq. (79), to the denominator One can easily extend the above MMCC(2,3)/CI approx-... 

It can be shown that orbital matrix elements that are off-diagonal in p spin-orbital indices only have non-vanishing matrix elements with 3.p,p+ 4>+z in particular the... 

When one imposes a definite order to the spin-orbital indices, i.e. i < i2 <. .. < ir and similarly for the j string, the factor T disappears. 

These equations show which excitation processes contribute at each excitation level. Thus, the quadruply excited configurations are generated by five distinct mechanisms, where, for instance, the disconnected term represents the independent interactions within two distinct pairs of electrons and the connected f4 terra describes the simultaneous interaction of four electrons. The disconnected terms represent interactions of product clusters within disjoint sets of electrons and vanish whenever two or more spin-orbital indices are identical. 

Higher excitations may be generated by an obvious extension of this scheme, avoiding duplicates and fixing the phase factors by applying suitable restrictions on the spin-orbital indices. An unspecified determinant may be written as... 

Thus, in the CISDt method, we construct wave functions ir) by including all singles and doubles from <>) and a small set of triples containing at least one active occupied and one active unoccupied spin-orbital indices. The Cl expansion coefficients defining wave functions are determined vari-... 

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 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 notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

Note that contributions from the secondary sector of the eigenvectors, Uf, do not appear in the residues, for the summation index, r, pertains to spin-orbitals only. 

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 ... 

First, we consider the case of two tlu electrons, later we will extend the results to more particles. Since the one electron spin-orbit coupling is negligible [13], we will work in the LS (Russel-Saunders) molecular approximation. Two electron basis ket vectors are denoted by a single index I ... 

Due to relativistic effects important for the inner shells of atoms, the deep ntf shells split in energy according to the j value from the spin-orbit coupling j = t + 1/2. Therefore, deep inner shells are classified by their nfj values. Frequently, in X-ray emission and Auger electron spectrometry, the shell index... 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

Here the summation index k runs over all common spin orbitals. 

Single excitations and I ] differ by one pair of spin orbitals i,j. Contributions come from the one-electron spin-orbit integral and three-index two-electron integrals... 

The summation index k runs over common spin orbitals. The second Coulomb-type integral (k(l)i(2) M so(l,2) k(l)j(2)) vanishes because is linear in... 

Only four-index two-electron integrals contribute to the spin-orbit coupling matrix element ... 

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. 

All of the foregoing equations and diagrams are in the spin-orbital formalisms, meaning that each matrix index carries the attributes of spatial orbital and spin. For instance, the MP2 energy [Eq. (2-29)] is the sum of the contributions from the following four spin combinations ... 

The summation shown here is over spin orbitals, but in practice it is carried out over spatial orbitals for which upper case indices are employed. The different spin cases which can arise are summarized in Table 1. The difference spin cases are distinguished by the index fi. 

Direct Cl methods often require an index vector which points to a list of all allowed excitations from a given iV-electron basis function. Using alpha and beta strings, the index vector need not be the length of the Cl vector—its size is dictated by the number of alpha or beta strings, which (for a full Cl) is approximately the square root of the number of determinants. This results from the fact that in determinant-based Cl, electrons in alpha spin-orbitals can be excited only to other alpha spin-orbitals, and electrons in beta spin-orbitals can be excited only to other beta spin-orbitals (because of the restriction to a single value of Ms). 

The (iV ) factor included in these expressions ensures that the determinants are normalized when the orbitals are normalized. Eq. (41) gives an explicit representation of the antisymmetrizer. This summation is over the N permutations of electron coordinates for a fixed orbital order, or equivalently, over the permutations of spin-orbital labels for a fixed order of electrons. The exponent Pp is the number of interchanges required to bring a particular permuted order of electron coordinates, or of spin-orbital labels, back to the original order. Different expansion terms are generated when different spin orbitals are employed in the determinant. For convenience, we will choose this spin-orbital basis to be the direct product of the set of n spatial orbitals and the set of spin factors a, / . A particular spin orbital of this form may be written as where r (= 1 to n) labels the spatial orbital and spin factor, or simply as (j), where the combined index r (= 1 to 2n) labels both the spatial and spin components. The notation used will be clear from the context. 

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