Computational methods electronic structure calculations

Chapter 1, Computational Models and Model Chemistries, provides an overview of the computational chemistry field and where electronic structure theory fits within it. It also discusses the general theoretical methods and procedures employed in electronic structure calculations (a more detailed treatment of the underlying quantum mechanical theory is given in Appendix A). 

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

The use of Eq. (5-10) to evaluate the reaction rate is characterised by the calculation of Hessians for a large number of points along the MEP which are required to locate the free energy maximum and also to evaluate the curvature required for evaluation of the transmission coefficient. In view of the associated computational expense, high-level electronic structure calculations are not feasible and alternative strategies, one of which is to use a semi-empirical method, are usually employed [81]. 

The basic idea underlying AIMD is to compute the forces acting on the nuclei by use of quantum mechanical DFT-based calculations. In the Car-Parrinello method [10], the electronic degrees of freedom (as described by the Kohn-Sham orbitals y/i(r)) are treated as dynamic classical variables. In this way, electronic-structure calculations are performed on-the-fly as the molecular dynamics trajectory is generated. Car and Parrinello specified system dynamics by postulating a classical Lagrangian ... 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

The Section on More Quantitive Aspects of Electronic Structure Calculations introduces many of the computational chemistry methods that are used to quantitatively evaluate molecular orbital and configuration mixing amplitudes. The Hartree-Fock self-consistent field (SCF), configuration interaction (Cl), multiconfigurational SCF (MCSCF), many-body and Mpller-Plesset perturbation theories,... 

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. 

There are (at least) two major opportunities for research by those interested in this topic. On the computational side, there is definite room for improvement in simulation methods. Right now none of the simulation approaches has the user friendliness that has brought electronic-structure calculation into the realm of routine applicability by nonspecialists. Nor has the field seen the development of the qualitative or semiquantitative models that did so much to make the results of molecular orbital calculations useful to organic chemists. On the experimental side, it will be obvious to the reader that the techniques for detecting the effects of nonstatistical dynamics are still very rudimentary and indirect. There is clearly room for creative scientists to come up with techniques whose results can give us more direct insight into these issues. 

C. J. Cramer, Essentials of Computational Chemistry, John Wiley Sons, Inc., New York, 2002. This monograph provides an authoritative, detailed, and very readable treatise on the methods that are currently used for performing electronic structure calculations, which are described briefly in Section 3. 

The use of computational chemistry to address issues relative to process design was discussed in an article. The need for efficient software for massively parallel architectures was described. Methods to predict the electronic structure of molecules are described for the molecular orbital and density functional theory approaches. Two examples of electronic stracture calculations are given. The first shows that one can now make extremely accurate predictions of the thermochemistry of small molecules if one carefully considers all of the details such as zero-point energies, core-valence corrections, and relativistic corrections. The second example shows how more approximate computational methods, still based on high level electronic structure calculations, can be used to address a complex waste processing problem at a nuclear production facility (Dixon and Feller, 1999). 

Computational chemistry methodology is finding increasing application to the design of new flavoring agents. This chapter surveys several useful techniques linear free energy relationships, quantitative structure-activity relationships, conformational analysis, electronic structure calculations, and statistical methods. Applications to the study of artificial sweeteners are described. 

In Section 1.1, parametric methods for improving tlie quality of correlated electronic-structure calculations were discussed in detail. Similarly, in Section 8.4.3, the mild parameterization of density functional methods to give maximal accuracy was described. Given that background, and the substantial data presented in diose earlier chapters, this section will only recapitulate in a rough categorical fashion tlie various approaches whose development was motivated by a desire to compute more accurate thermochemical quantities. 

Also pure density-functional methods combined with plane-wave basis sets and ultrasoft pseudopotentials [58] were used in our studies of extended systems [59]. The computational efficiency of these methods enables larger systems and to some extent dynamical processes to be studied. Generalized-gradient approximation (GGA) or spin-polarized GGA DFT functionals [60, 61] were employed in the electronic structure calculations. 

J. Kohanoff, Electronic Structure Calculations for Solids and Molecules Theory and Computational Methods, Cambridge, New York, 2006. 

In the above formula, Q is the nuclear coordinate, p, and I/r are the ground state and excited electronic terms. Here Kv is provided through the traditional Rayleigh-Schrodinger perturbation formula and K0 have an electrostatic meaning. This expression will be called traditional approach, which has, in principle, quantum correctness, but requires some amendments when different particular approaches of electronic structure calculation are employed (see the Bersuker s work in this volume). In the traditional formalism the vibronic constants P0 dH/dQ Pr) can be tackled with the electric field integrals at nuclei, while the K0 is ultimately related with electric field gradients. Computationally, these are easy to evaluate but the literally use of equations (1) and (2) definitions does not recover the total curvature computed by the ab initio method at hand. 

---