Hamiltonian operators molecular properties

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

Having considered the general expressions for first- and second-order molecular properties, we now restrict ourselves to properties associated with the application of static uniform external electric and magnetic fields. For such perturbations, the Hamiltonian operator may be written in the manner (in atomic units)... 

In this paper, both theories will be briefly reviewed presenting their differences. From these comparisons, the Non-rigid Molecule Group (NRG) will be stricktly defined as the complete set of the molecular conversion operations which commute with a given Hamiltonian operator [21]. The operations of such a set may be written either in terms of permutations and permutations-inversions, just as in the Longuet-Higgins formalism, or either in terms of physical operations just as in the formalism of Altmann. But, the order, the structure, the symmetry properties of the group will depend exclusively on the Hamiltonian operator considered. 

Molecular properties can accordingly be defined as energy derivatives at zero perturbation strength [1]. Such energy derivatives can be evaluated numerically or analytically. The numerical approach consists of adding the relevant perturbation operator(s) to the zeroth order Hamiltonian and then calculate the energy... 

There are two sides to the deperturbation process the Hamiltonian model and the partially assigned and analyzed spectrum. The molecular Hamiltonian operator must be organized in a way that, of the infinite number of molecular bound states and continua, emphasis is placed on the finite subset of levels sampled in the spectrum under analysis. A computational model is constructed in which the unknown information about the relevant states is represented by a set of adjustable parameters that may be systematically varied, until an acceptable match between calculated and observed properties is obtained. 

In molecular property calculations the same mutual interplay of electron correlation, relativity and perturbation operators (e.g. external fields) occurs. For light until medium atoms relativistic contributions were often accounted for by perturbation theory facilitating quasirela-tivistic approximations to the Dirac-Hamiltonian [114-117]. It is well-known that operators like the Breit-Pauli Hamiltonian are plagued by essential singularities and therefore are not to be used in variational procedures. It can therefore be expected that for heavier elements per-turbational inclusion of relativity will eventually become inadequate and that one has to start from a scheme where relativitiy is included from the beginning. Nevertheless very efficient approximations to the Dirac equation in two-component form exist and will be discussed further below in combination with their relevance for EFG calculations. In order to calculate the different contributions to a first-order property as the EFG, Kello and Sadlej devised a multiple perturbation scheme [118] in which a first-order property is expanded as... 

In fact, the wave operator 2 can be used to evaluate both, the effective Hamiltonian Heff as well as any other atomic or molecular property of interest. Since Hgff is defined only within the model space M, however, it is often more convenient for open-shell systems to evaluate the matrix elements of = PHi2 " P (or of any other interaction operators) with regard to the functions 4>a) M, instead of deriving an explicit representation of the effective Hamiltonian. From the diag-onalization of the Hamiltonian matrix, calculated up to the order n in the wave operator, we then immediately obtain the energies up to the order n + 1, as... 

The electronic wave function of an n-electron molecule depends on 3ra spatial and n spin coordinates. Since the Hamiltonian operator (15.10) contains only one- and two-electron spatial terms, the molecular energy can be written in terms of integrals involving only six spatial coordinates (Problem 15.82). In a sense, the wave function of a many-electron molecule contains more information than is needed and is lacking in direct physical significance. This has prompted the search for functions that involve fewer variables than the wave function and that can be used to calculate the energy and other properties. 

The first step in the application of symmetry to molecular properties is therefore to recognize and organize all of the symmetry elements that the molecule possesses. A symmetry element is an imaginary point, line, or plane in the molecule about which a symmetry operation is performed. An operator is a symbol that tells you to do something to whatever follows it. Thus, for example, the Hamiltonian operator is the sum of the partial differential equations relating to the kinetic and... 

Similar to nonrelativistic Hartree-Fock theory, the Dirac-Roothaan Eqs. (10.61) are solved iteratively until self-consistency is reached. However, because of the properties of the one-electron Dirac Hamiltonian entering the Fock operator, molecular spinors representing unphysical negative-energy states (recall section 5.5) show up in this procedure. As many of these negative-continuum... 

In this chapter, we shall now come back to the question how physical observables are associated with proper operator descriptions, which has already been addressed in section 4.3. All preceding chapters dealt with the proper construction of Hamiltonians for the calculation of energies and wave functions of many electron systems. Here, we shall now transfer this knowledge to the construction of relativistic expressions for first-principles calculations of molecular properties for many-electron systems. The basic guideline for this is the fact that all molecular properties can be expressed as total electronic energy derivatives. 

In general, they can all be properly dealt with in the framework of perturbation (response) theory. According to the discussion in section 5.4, we may add external electromagnetic fields acting on individual electrons to the one-electron terms in the Hamiltonian of Eq. (8.66). Fields produced by other electrons, so that contributions to the one- and two-electron interaction operators in Eq. (8.66) arise, are not of this kind as they are considered to be internal and are properly accounted for in the Breit (section 8.1) or Breit-Pauli Hamiltonians (section 13.2). Although the extemal-field-free Breit-Pauli Hamiltonian comprises all internal interactions, such as spin-spin and spin-other-orbit terms, they may nevertheless also be considered as a perturbation in molecular property calculations. While our derivation of the Breit-Pauli Hamiltonian did not include additional external fields (such as the magnetic field applied in magnetic resonance spectroscopies), we now need to consider these fields as well. 

At this stage we should add the missing extemal-field-dependent operators to the Breit-Pauli Hamiltonian reviewed already by Bethe [72]. By contrast to what follows, these terms are also derived in the spirit of the ill-defined Foldy-Wouthuysen expansion in powers of 1 /c. However, since the molecular property calculation is carried out in a perturbation theory anyhow, we may utilize the complete field-dependent Breit-Pauli Hamiltonian in such calculations. 

---