Operator spin-orbit

The HF [31] equations = e.cj). possess solutions for the spin orbitals in T (the occupied spin orbitals) as well as for orbitals not occupied in F (the virtual spin orbitals) because the operator is Flennitian. Only the ( ). occupied in F appear in the Coulomb and exchange potentials of the Fock operator. 

As fonnulated above, the FIF equations yield orbitals that do not guarantee that F has proper spin symmetry. To illustrate, consider an open-shell system such as the lithium atom. If Isa, IsP, and 2sa spin orbitals are chosen to appear in F, the Fock operator will be... 

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 ... 

Now, we add to (1) the operator describing the spin-orbit (SO) coupling, so that our model Hamiltonian becomes... 

We shall consider only the leading part of the spin-orbit operator assumed in the phenomenological fonn... 

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

I hi5 expression can be tidied up by introducing three operators that represent the contribu-tio.is to the energy of the spin orbital Xi in the frozen system ... 

In particular, within the orbital model of eleetronie strueture (whieh is developed more systematieally in Seetion 6), one ean not eonstruet trial waveflmetions whieh are simple spin-orbital produets (i.e., an orbital multiplied by an a or P spin funetion for eaeh eleetron) sueh as lsalsP2sa2sP2pia2poa. Sueh spin-orbital produet funetions must be made permutationally antisymmetrie if the N-eleetron trial funetion is to be properly antisymmetrie. This ean be aeeomplished for any sueh produet wavefunetion by applying the following antisymmetrizer operator ... 

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,... 

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. 

The first set of equations govern the Cj amplitudes and are ealled the Cl- seeular equations. The seeond set determine the LCAQ-MQ eoeffieients of the spin-orbitals ([ij and are ealled the Foek equations. The Foek operator F is given in terms of the one- and two-eleetron operators in H itself as well as the so-ealled one- and two-eleetron density matriees Yij and Fi j i whieh are defined below. These density matriees refleet the averaged oeeupaneies of the various spin orbitals in the CSFs of P. The resultant expression for F is ... 

For sueh a funetion, the CI part of the energy minimization is absent (the elassie papers in whieh the SCF equations for elosed- and open-shell systems are treated are C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951) 32, 179 (I960)) and the density matriees simplify greatly beeause only one spin-orbital oeeupaney is operative. In this ease, the orbital optimization eonditions reduee to ... 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

---