Hamiltonian operators electronic structure methods

This effective interaction Hamiltonian operator is difficult to evaluate as it requires the knowledge of the transition dipole moments between the essential states and all the non-essential states. For a general molecular system, there is an infinity of such states, including bound states but also the various continua. In practice however, one can further approximate the second term of Eq. (6.51) by neglecting the (ro)vibrational structure of the non-essential electronic states. This term can then be expressed as a function of static electronic polarizabilities, that can be computed using standard electronic structure methods available in the major quantum chemistry program packages. 

We recall that the Hamiltonian, O Eq. 11.2, does not involve the electron spin. Spin-restricted implementations of electronic structure methods, therefore, only optimize the wave function parameters of the same spin symmetry as the reference wave function. Since operators that change the spin symmetry are known not to contribute to the ground-state energy (at least in the nonrelativistic picture in which spin-orbit effects are ignored), such spin-breaking wave function parameters are left imdefined. This has the consequence that average values of... 

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. 

On the other hand, ab initio (meaning from the beginning in Latin) methods use a correct Hamiltonian operator, which includes kinetic energy of the electrons, attractions between electrons and nuclei, and repulsions between electrons and those between nuclei, to calculate all integrals without making use of any experimental data other than the values of the fundamental constants. An example of these methods is the self-consistent field (SCF) method first introduced by D. R. Hartree and V. Fock in the 1920s. This method was briefly described in Chapter 2, in connection with the atomic structure calculations. Before proceeding further, it should be mentioned that ab initio does not mean exact or totally correct. This is because, as we have seen in the SCF treatment, approximations are still made in ab initio methods. 

The method discussed here for the inclusion of relativistic effects in molecular electronic structure calculations is grounded in the Dirac-Fock approximation for atomic wave functions (29). The premise is that the major relativistic effects of the Dirac Hamiltonian are manifested in the core region, involving the core electrons, and that these effects propagate to the valence electrons. In addition, there are direct relativistic effects on valence electrons penetrating into the core region. Insofar as this is true, the valence electrons can be treated using a nonrelativistic Hamiltonian to which is added an operator, the relativistic effective core potential (REP). The REP formally, incorporates relativistic effects due to core electrons and to interactions of valence electrons with core electrons in an internally consistent way. 

It should be emphasized that OEMO theory does not treat explicitly interelectronic repulsions, which are reproduced by the two electron part of a complete hamiltonian operator, as well as internuclear repulsions. These effects are partially accounted for by virtue of the empirical evaluation of matrix elements in the OEMO method and will be grouped undo-the heading steric effects . It is obvious that steric effects will tend to favor uncongested structures. It is then apparent that the OEMO theory will lead to incorrect predictions when steric effects become a dominant influence. 

The CEO computation of electronic structure starts with molecular geometry, optimized using standard quantum chemical methods, or obtained from experimental X-ray diffraction or NMR data. For excited-state calculations, we usually use the INDO/S semiempirical Hamiltonian model (Section IIA) generated by the ZINDO code " however, other model Hamiltonians may be employed as well. The next step is to calculate the Hartree— Fock (HE) ground state density matrix. This density matrix and the Hamiltonian are the Input Into the CEO calculation. Solving the TDHE equation of motion (Appendix A) Involves the diagonalization of the Liouville operator (Section IIB) which is efficiently performed using Kiylov-space techniques e.g., IDSMA (Appendix C), Lanczos (Appendix D), or... 

Current relativistic electronic structure theory is now in a mature and well-developed state. We are in possession of sufficiently detailed knowledge on relativistic approximations and relativistic Hamiltonian operators which will be demonstrated in the course of this book. Once a relativistic Hamiltonian has been chosen, the electronic wave function can be constructed using methods well known from nonrelativistic quantum chemistry, and the calculation of molecular properties can be performed in close analogy to the standard nonrelativistic framework. In addition, the derivation and efficient implementation of quantum chemical methods based on (quasi-)relativistic Hamiltonians have facilitated a very large amount of computational studies in heavy element chemistry over the last two decades. Relativistic effects are now well understood, and many problems in contemporary relativistic quantum chemistry are technical rather than fundamental in nature. 

The relativistic Hamilton operator for an electron can be derived, using the correspondence principle, from its relativistic classical Hamiltonian and this leads to the one-electron Dirac equation, which does contain spin operators. From the one-electron Dirac equation it seems trivial to define a many-electron relativistic equation, but the generalization to more electrons is less straightforward than in the non-relativistic case, because the electron-electron interaction is not unambiguously defined. The non-relativistic Coulomb interaction is often used as a reasonable first approximation. The relativistic treatment of atoms and molecules based on the many-electron Dirac equation leads to so-called four-component methods. The name stems from the fact that the electronic wave functions consist of four instead of two components. When the couplings between spin and orbital angular moment are comparable to the electron-electron interactions this is the preferred way to explain the electronic structure of the lowest states. 

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