Spin-independent matrices

In Eq. (84), the a- and jS-dependences of have been retained only to emphasize that it is related to these SOs. We recall that If is a spin-independent matrix as a consequence of the antisymmetric properties of cumulant... 

Here nig denotes the sign of the spin projection (it takes two values, +1 and — 1). By taking into account the cumulant properties, Eqs. (10)-(13) with replaced by F, it can be shown that A must be a real symmetric matrix (A " = A j) with no unique diagonal elements, whereas 11 is a spin-independent (11 = =... 

We assume further that the ON of the open shell p is always one (tip = 1). This assumption is trivial for a doublet, but it is more restrictive for higher multi-plet with a corresponding underestimation of the energy. Remember that matrix elements of are nonvanishing only if aU its labels refer to partially occupied NOs therefore A = 0 and II = 0 if we consider a cumulant made up of at least one open-shell level. Since A and II refer only to closed shells, we consider them spin-independent. The sum rule, Eq. (89), becomes... 

Let us first become familiar with the spin-free (sf) case we have a pair of Slater determinants interacting via (i.e., a sum of spin-independent one-and two-electron operators, h and 12). Their matrix element is given by (Sla-ter-Condon rules) ... 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

Older versions of SOCI programs are very I/O intensive because they used to store the Hamiltonian matrix on disk and read it in every iteration step.52,141 Integral-driven direct methods for spin-orbit coupling came up in the mid 1980s123,142 following the original fomulation of direct Cl methods for spin-independent Hamiltonians.143-146 Modern direct SOCI programs can easily handle several million determinants.108,147-151... 

In spin-free quantum chemistry, matrix elements of a spin-independent... 

The matrix defined in Eq.(47) represents a general, non-relativistic spin-independent TV-electron Hamiltonian given by Eq.(2) in a specifically defined model space. The one-electron orbital space is spanned by N orthonormal localized orbitals 4>j, j = 1,2,..., TV and the TV-electron orbital space is one-dimensional with the basis function... 

Island counting is well suited for normalization or for matrix elements of spin-independent properties such as dipole moments, transition dipoles or charge-charge... 

Next we can dispose of the matrix elements of the one-electron operator Hy,Eq. (3-3 b) the kinetic energy operator, the electron-nuclear attraction potential arising from the metal nucleus, and the spin-independent relativistic terms have spherical symmetry and can be treated through the definition of the basis orbitals ip, Eq. (3-11). The spin-... 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

Since the operators f and f2 occur only at the level of the calculation of the spatial spin-orbit integrals over atomic orbitals, Breit-Pauli spin-orbit coupling operators and DKH spin-orbit coupling operators can be discussed on the same footing as far as their matrix elements between multi-electron wave functions are concerned. These terms constitute, by definition, the spin-orbit interaction part of the operator H+ (Hess etal. 1995). The spin-independent terms characteristic of relativistic kinematics define the scalar relativistic part of the operator, and terms with more than one cr matrix (not considered here) contribute to spin-spin coupling phenomena. 

For A > 0 states, second-order spin-orbit effects cause two types of J-independent level shifts proportional to AE (the form of the diagonal spin-orbit matrix element) or to 3E2 — S(S+1) (the form of the diagonal spin-spin matrix element). Second-order spin-orbit effects are unobservable for E and 2E states because they result in no new level splittings or changes in existing separations. For E states with S > 3, second-order spin-orbit effects are observable only as shifts similar to the form of the spin-spin interaction. See Levy (1973) for a demonstration of this fact. 

A model hamiltonian should have the structure of the full hamiltonian, but could in principle have terms consisting of higher order products of annihilation and creation operators. Here we limit considerations to such operators that contain a one-electron part and an electron-electron interaction part. The number of independent matrix elements can be considerably reduced by symmetry considerations and by requiring compatibility with other operator representatives. It is clear that the form of the spectral density requires that the hamiltonian commutes with the total orbital angular momentum and with various spin operators. These are given in the limited basis as... 

The matrix elements of r, and a are of rank-one, r is the exact analogue of the isospin-changing El operator and the conserved-vector-current (CVC) theorem can be invoked to relate the matrix element of a to that of r. Further, [r, spin-dipole operator. The helicity operator 75 is the time-like component of the rank-zero axial current - also known as the axial charge - discussed in Section 1. We thus have five independent matrix elements, two each of rank zero (RO) and rank one (Rl) and one of rank two (R2) which we denote as... 

Substitution (101) appears to be the simplest way in which spin-dependent phenomena can be incorporated into the theory. The operator (100) commutes with S and is thus consistent with effects produced by such spin-independent interactions as the Coulomb and crystal-field terms in the Hamiltonian. Moreover, the effect of (101) turns out to be equivalent to adding to each reduced matrix element of F " a part (proportional to c,) that involves the reduced matrix element of the double tensor where = s (Judd 1977b). Such double tensors are straightforward to evaluate furthermore the proportionality = /4 holds for all terms of... 

In an actual process spin-dependent interactions are often unimportant and therefore the involved quantities depend only on the orbital angular momenta L rather than on J. For a comparison with experimental data one therefore has to consider what the quantities R, p/, and Fj are in such cases. First of all the autoionization process itself can often be regarded as spin independent, at least in those cases in which the normal autoionization due to Coulomb interaction between the electrons is allowed. In that case the reduced matrix elements and thus the coefiicients R can be expressed in terms of L-dependent matrix elements by ... 

If the autoionization process only depends on L rather than on / the excited atom has to be characterized by the L-dependent tensor components PkqiL) in place of the /-dependent ones. This makes sense anyway, since the excitation processes in heavy particle collisions are mostly spin independent. Therefore the spin can be regarded as spectator during the excitation process, and at time t = 0, when the excitation takes place, the density matrix factorizes into an L-dependent part and a spin-dependent one which is diagonal. The pkq L,T = 0) therefore characterize the excitation process and are the quantities that one would like to know in order to obtain information about excitation mechanisms. 

The total density matrix is used to get the expectation value of one-electron operators that are spin independent... 

In the nonrelativistic case, with a and p strings. Mg = Ms- Because the nonrelativistic operators are spin-independent, the Hamiltonian matrix is blocked by Ms- This block diagonalization of the Hamiltonian matrix does not persist in the relativistic case, and in the absence of any point group symmetry, the N-particle basis extends to all Mg values. 

Since Hi is spin-independent, the first matrix element vanishes unless aM) and bM") have identical spin eigenvalues, and then takes the A/-independent value... 

In addition, using restricted Hartree-Fock orbitals, jp2 m l is the matrix element over spatial orbitals assumed to the independent of spin. The matrix M describes multipair excitations. In M = 0, Eqs. (2.33) are separable. Then their diagonal elements yield... 

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