Spin-orbit matrix elements, determination

To get the leading terms of ( j H 1 d>2), we will consider only the matrix elements between the determinants that differ by only one orbital, after permutations to put them in maximum spin-orbital correspondence, that is, ( af // c/ ) and ( ab h cb ). The terms that are to the power of two, (35 or 52, are neglected. 

Here, grs is a parameter that is quantified either from experimental data, or is calculated by an ab initio method as one-half of the singlet-triplet excitation energy gap of the r—s bond. In terms of the qualitative theory in Chapter 3, grs is therefore identical to the key quantity —2(3 5 - This empirical quantity incorporates the effect of the ionic components of the bond, albeit in an implicit way. (c) The Hamiltonian matrix element between two determinants differing by one spin permutation between orbitals r and s is equal to grs. Only close neighbor grs elements are taken into account all other off-diagonal matrix elements are set to zero. An example of a Hamiltonian matrix is illustrated in Scheme 8.1 for 1,3-butadiene. 

Since the many-electron wave function can be expanded in a linear combination of Slater determinants, its matrix element with a spin-orbit coupling operator of the form of Equation (3.6) can be expressed as a sum of matrix elements of the operator between Slater determinants. For a matrix element between Slater determinants which differ in exactly one spin orbital (i.e. which are singly excited from i — a with respect to each other), the matrix element is... 

The absolute sign of an off-diagonal matrix element cannot be determined, since it depends arbitrarily on the chosen phase for the determinantal wave function, namely, on the order of the spin-orbitals. However, the relative sign of two off-diagonal matrix elements can often be determined experimentally. Thus, care must be taken to define the phases of the wavefunctions consistently. 

The two-electron spin-orbit integrals contribute to the spin-orbit matrix element between Slater determinants which are singly or double excited relative to one another. The matrix elements between singly excited determinants can, just like in the Hartree-Fock equations, be written as a pseudo one-electron integral. One of the key aspects of a mean-field theory is to neglect interactions between double excited states and to include all two-electron integrals in pseudo one-electron integrals. 

McWeeny, R. and Sutcliffe, B. T., Methods of Molecular Quantum Mechanics 2nd ed., Academic Press, New York, 1976. Discusses the rules for evaluating matrix elements between Slater determinants formed from nonorthogonal spin orbitals. This book also contains a concise introduction to methods for obtaining spin eigenfunctions. 

The Hamiltonian matrix element between two determinants differing by one spin permutation between orbitals r and s is equal to Any other off-diagonal matrix elements are set to zero (see Scheme 12). 

Since all determinants, created by the same configuration, with a given S are related by double and higher excitations (5), determinants Dm and Dt can interact only if exactly two open-shell orbitals a and b of opposite spin in Dm have inverse spins in A- Taking into account that the Hamiltonian matrix element between such determinants is -Kab, on the basis of Eqs. (6), (32) and (33), one has... 

Before proceeding to determine the form of the operators in second quantization, we recall that the matrix elements between Slater determinants depend on the spatial form of the spin orbitals. Since the ON vectors are independent of the spatial form of spin orbitals, we conclude that the second-quantization f )eratws - in contrast to their first-quantization counterparts - must depend on the spatial form of the spin orbitals. 

Other examples of two-electron operators are the two-electron part of the spin-orbit operator and the mass-polarization operator. A two-electron operator gives nonvanishing matrix elements between Slater determinants if the determinants contain at least two electrons and if they differ in the occupations of at most two pairs of electrons. The second-quantization representation of a two-electron operator therefore has the structure... 

The so-ealled Slater-Condon rules express the matrix elements of any one-eleetron (F) plus two-eleetron (G) additive operator between pairs of antisymmetrized spin-orbital produets that have been arranged (by permuting spin-orbital ordering) to be in so-ealled maximal eoineidenee. Onee in this order, the matrix elements between two sueh Slater determinants (labelled >and are summarized as follows ... 

As a first step in applying these rules, one must examine > and > and determine by how many (if any) spin-orbitals > and > differ. In so doing, one may have to reorder the spin-orbitals in one of the determinants to aehieve maximal eoineidenee with those in the other determinant it is essential to keep traek of the number of permutations ( Np) that one makes in aehieving maximal eoineidenee. The results of the Slater-Condon rules given below are then multiplied by (-l) p to obtain the matrix elements between the original > and >. The final result does not depend on whether one ehooses to permute ... 

All sueh matrix elements, for any one- and/or two-eleetron operator ean be expressed in terms of one- or two-eleetron integrals over the spin-orbitals that appear in the determinants. 

When the states P1 and P2 are described as linear combinations of CSFs as introduced earlier ( Fi = Zk CiKK), these matrix elements can be expressed in terms of CSF-based matrix elements < K I eri IOl >. The fact that the electric dipole operator is a one-electron operator, in combination with the SC rules, guarantees that only states for which the dominant determinants differ by at most a single spin-orbital (i.e., those which are "singly excited") can be connected via electric dipole transitions through first order (i.e., in a one-photon transition to which the < Fi Ii eri F2 > matrix elements pertain). It is for this reason that light with energy adequate to ionize or excite deep core electrons in atoms or molecules usually causes such ionization or excitation rather than double ionization or excitation of valence-level electrons the latter are two-electron events. 

As a result, the exaet CC equations are quartic equations for the ti , ti gte. amplitudes. Although it is a rather formidable task to evaluate all of the eommutator matrix elements appearing in the above CC equations, it ean be and has been done (the referenees given above to Purvis and Bartlett are espeeially relevant in this eontext). The result is to express eaeh sueh matrix element, via the Slater-Condon rules, in terms of one- and two-eleetron integrals over the spin-orbitals used in determining , ineluding those in itself and the Virtual orbitals not in . 

In the unrestricted Hartree-Fock method, a single-determinant wave function is used with different molecular orbitals for a and jS spins, and the eigenvalue problem is solved with separate F and F matrices. With the zero differential overlap approximation, the F matrix elements (25) become... 

EOMCCSD(T) calculations, while facilitating the open-shell implementation of the CR-EOMCCSD(T) method employing the restricted open-shell Hartree-Fock (ROHE) orbitals [59]. Indeed, the use of spin-orbital energy differences (ca -f -f Cc — — ej — e ) instead of the complete form of the diagonal matrix elements of involving triply excited determinants to... 

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

With the further condition that the spin-orbitals are orthogonal the special cases, Slater s rules, for matrix elements between determinants are obtained from this formula by inspection. The general formula can be written... 

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