Spin Hamiltonians calculation from molecular orbitals

The ESR spin Hamiltonian parameters have been calculated using 77-HMO and tt-SCF MO coefficients in the molecular orbitals as well as those from nonempirical wavefunctions. It may be seen that, in general, calculated values are in good agreement with experimental data (Table XXXI). 

S =. As pointed out by Ohno et al. 45) the molecular orbital calculation for the ferric case lacks internal consistency. Hence we turn to the information based on ESR experiments. As has been discussed in Ch. VII, the zero field splitting parameter which appears in the spin Hamiltonian (Eq. 131) may be estimated from the orbital energies derived from ESR data on the low spin hemoglobin azide (Fig. 23). Using the values... 

FVom the above discussion, it follows that the construction of parameterized hamiltonians for molecular orbital calculations may lead to certain operator relationships being violated in a limited basis. It is also clear that some of these operator relations can be restored at the expense of introducing various approximations in the evaluation of integrals. Atomic parameters may be derived from consideration of the separated atoms limit, while interatomic parameters are commonly associated with overlap integrals and possibly other functions of the interatomic distance. For instance, it is often assumed that when r is a spin orbital on atom A and s is one on atom B, a suitable form for the hopping term is... 

If the Hartree-Fock equations associated with the valence pseudo-Hamiltonian (167) are solved with extended basis sets, then all the above F are almost basis-set-independent. At the present time, and for practical reasons, most of the ab initio valence-only molecular calculations use coreless pseudo-orbitals. The reliability of this approach is still a matter of discussion. Obviously the nodal structure is important for computing observable quantities such as the diamagnetic susceptibility which implies an operator proportional to 1/r. From the computational point of view, it is always easy to recover the nodal structure of coreless valence pseudo-orbitals by orthogonalizing the valence molecular orbitals to the core orbitals. This procedure has led to very accurate results for several internal observables in comparison with all-electron results. The problem of the shape of the pseudoorbitals in the core region is also important in relativity. For heavy atoms, the valence electrons possess high instantaneous velocities near the nuclei. Schwarz has recently investigated the compatibility between the internal structure of valence orbitals and the representations of operators such as the spin-orbit which vary as 1/r near the nucleus. ... 

We will begin this chapter by constructing determinantal trial functions from the Hartree-Fock molecular orbitals, obtained by solving Roothaan s equations. It will prove convenient to describe the possible N-electron functions by specifying how they differ from the Hartree-Fock wave function Fo Wave functions that differ from Tq by w spin orbitals are called n-tuply excited determinants. We then consider the structure of the full Cl matrix, which is simply the Hamiltonian matrix in the basis of all possible N-electron functions formed by replacing none, one, two,... all the way up to N spin orbitals in Section 4.2 we consider various approximations to the full Cl matrix obtained by truncating the many-electron trial function at some excitation level. In particular, we discuss, in some detail, a form of truncated Cl in which the trial function contains determinants which differ from To by at most two spin orbitals. Such a calculation is referred to as singly and doubly excited Cl (SDCI). 

The two-electron operator is given in the nuclear frame and not in the reference of either electron. The spin-orbit coupling due to the relative motion of elecrons therefore splits into two parts The total interaction is the coupling of the spin of a selected reference electron with the magnetic field induced by a second electron. The spin-same orbit (SSO) and spin-other orbit (SOO) contributions arise from the motion of the reference electron and the other electron, respectively, relative to the nuclear frame and are carried by the Coulomb and Gaunt terms, respectively. For most molecular application it suffices to include the Coulomb term only, thus defining the Dirac-Coulomb Hamiltonian, but for the accurate calculation of molecular spectra the Gaunt term should be included as well. 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

Hay and Wadt (1985a, b) have published ECPs which are in form identical to the averaged RECPs of Christiansen, Ermler and co-workers. However, there are differences. First, the Hay-Wadt potentials are derived from the Cowan-GriflSn adaptation of the Breit-Pauli Hamiltonian into a variational computation of the atomic wave-function. From these solutions the ECPs are generated. It should be noted that the spin-orbit coupling is not included in the Hay-Wadt ECPs. Consequently, molecular calculations done using these ECPs would not include spin-orbit coupling. 

Semiempirical molecular quantum-mechanical methods use a simpler Hamiltonian than the correct molecular Hamiltonian and use parameters whose values are adjusted to fit experimental data or the results of ab initio calculations. An example is the Hiickel MO treatment of conjugated hydrocarbons (Section 17.2), which uses a one-electron Hamiltonian and takes the bond integrals as adjustable parameters rather than quantities to be calculated theoretically. In contrast, an ab initio (or first principles) calculation uses the correct Hamiltonian and does not nse experimental data other than the values of the fundamental physical constants. A Hartree-Fock SCF calculation seeks the antisynunetrized product d> of one-electron functions that minimizes / dr, where H is the true Hamiltonian and is thns an ab initio calcnlation. (Ab initio is Latin for from the beginning and indicates a calculation based on fundamental principles.) The term ab initio should not be interpreted to mean 100% correct. An ab initio SCF MO calculation uses the approximation of taking as an antisynunetrized product of one-electron spin-orbitals and uses a finite (and hence incomplete) basis set. 

The Cowan-Griffin Hamiltonian was developed for spin-free relativistic atomic calculations (Cowan and Griffin 1976). However, it has also found some use as a starting point for developing spin-free relativistic Hamiltonians for molecular application. Here, we show the form of this operator, and the associated spin-orbit correction. For atoms, the Cowan-Griffin Hamiltonian follows directly from the radial form of the atomic Dirac equation (7.29), which may be given as... 

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