Electronic-Structure Calculations

After an introduction to methods of electronic structure calculations, we review how recent trends translate into the description of magnetic nanostructures. Among the considered structures are nanowires, small particles, surfaces and interfaces, and multilayers, and emphasis is on magnetic properties such as moment and magnetization, interatomic exchange, and anisotropy. 

Electronic structure methods for studies of nanostructures can be divided broadly into supercell methods and real-space methods. Supercell methods use standard k-space electronic structure techniques separating periodically repeated nanostructures by distances large enough to neglect their interactions. Direct space methods do not need to use periodic boundary conditions. Various electronic structure methods are developed and applied using both approaches. In this section we will shortly discuss few popular but powerful electronic structure methods the pseudopotential method, linear muffin-tin orbital and related methods, and tight-binding methods. 

The pseudopotential method relies on the separation (in both energy and space) of electrons into core and valence electrons and implies that most physical and chemical properties of materials are determined by valence electrons in the interstitial region. One can therefore combine the full ionic potential with that of the core electrons to give an effective potential (called the pseudopotential), which acts on the valence electrons only. On top of this, one can also remove the rapid oscillations of the valence wavefunctions inside the core region such that the resulting wavefunction and potential are smooth. 

Beyond a chosen cutoff radius, the all-electron and pseudofunctions (potential and wavefunction) are identical, while inside the core region both the pseudopotential and pseudowavefunction are smoothly varying. After the construction of these pseudofunctions for a single atom and ensuring that their scattering properties are almost identical to those of the all-electron functions, they can be used in any chemical environment. 

The pseudopotential method has various advantages. Eliminating the core electrons from the problem reduces the number of particles that must be considered in the Kohn-Sham (KS) equations for the effective one-electron potential. For example, a pseudopotential calculation for bulk silicon (with 10 core and 4 valence electrons) requires the calculation of 4 occupied bands at each k-point, while an all-electron approach would require the calculation of 14 occupied bands. More importantly, the smooth spatial variation of the pseudopotential and pseudowavefunction allows the use of computationally convenient and unbiased basis, such as plane wave basis sets or grids in space. 

The present section contains some results from calculations of orbital energies ej, total energies Ej, and 0-F overlap populations (o.p.). Theoretical results for molecular properties, such as dipole moment, ionization potential, and geometric structure, are mentioned in the respective sections. 

Data have been taken mainly from ab initio self-consistent-field (SCF) calculations. The thermochemical stability of OFg can, however, only be accounted for by the inclusion of correlation effects [1]. 

The lowest ab initio SCF energy obtained so far, = - 273.5594 a.u., has apparently been calculated [2] at the optimized geometry with a polarized double or triple zeta basis of Slater -type orbitals (STO). The Hartree-Fock (HF) limit of the total energy has been estimated [10] to be EV = -273.68 a.u. A slightly higher estimate is due to Rothenberg and Schaefer [8]. 

Additional ab initio SCF energies are collected in the following table. The orbitals, for which E, were given in the original publications, are indicated as well as the geometry and the basis set (GTO = Gaussian-type orbital, DZ or TZ = double or triple zeta, P = polarization)  

For ab initio SCF calculations with minimal basis sets, see also [15 to 17]. 

In general, a molecule with N atoms has 3N — 6 internal degrees of freedom denoted by R = (ri,r-i.However, if the molecule is linear it has 3N — 5 internal degrees of freedom. The potential energy surfaces are functions of the in- 

If several electronically excited states are relevant for describing the photodissociation then one or more of the Rydberg orbitals of the molecule must be included in the (CAS) [13], As the number of orbitals and electrons increases in the CAS, the computational time increases dramatically. In order to obtain accurate potential energy surfaces for the excited electronic states, one must include diffuse functions in the basis set [4], For heavier atoms, a relativistic effective core potential (ECP) can be used to treat the scalar relativistic effects. The ECP basis sets have been developed by several research groups [15,16] and have been implemented in most of the standard electronic structure programs. 

The sudden changes in the adiabatic wavefunctions near avoided crossings make it more convenient to use diabatic potential energy surfaces when simulating photodissociation dynamics. The adiabatic potentials, usually constructed from electronic structure calculation data, should therefore be transformed to diabatic potentials. The adiabatic-diabatic transformation yields diabatic states for which the derivative couplings above approximately vanish. The diabatic potential energy surfaces are obtained from the adiabatic ones by a unitary orthogonal transformation [22,23] 

Photodissociation involves electronic transitions initiated by the absorption of light. The key molecular property that mediates the interaction with light is the transition dipole moment. The electronic transition dipole moment between the jth and kth electronic state is defined by the integral over the electronic degrees of freedom for the operator where the dipole moment is sandwiched between the two electronic wave functions given by 

The electronic transition dipole moment depends on the nuclear coordinates. Certain transitions are forbidden by symmetry, for example in CO2 and N2O. 

In general, only small molecules, usually diatomics, have been studied with four-component methods. Often, correlation effects have not yet been taken into account. Those larger molecules, which have also been studied to some extent, exhibit high symmetry like Oh or 7d consisting of only two symmetry-inequivalent atoms. Therefore, hydrides, oxides and halides are by far the most extensively studied molecules. 

One purpose of these calculations is to understand the effect of a four-component treatment for different types of molecules to evaluate the reliability of more approximate treatments like two-component or one-component methods. In other words, those cases must be identified where only four-component calculations yield sufficiently accurate results. In all other cases, more approximate methods, which do 

For molecules the evaluation of the Breit correction to the Coulomb-type electron-electron interaction operator becomes computationally highly demanding and cannot be routinely evaluated, not even on the Dirac-Fock level. To test the significance of the Breit interaction, the Gaunt term is evaluated as a first-order perturbation. It turned out that it can be neglected in most cases as can be seen from the DF 4- Bmag calculations cited in Table 2.1. 

The UF6 molecule has also been studied extensively using a more elaborate method, namely configuration interaction, to assign the experimental photoelectron spectrum (de Jong and Nieuwpoort 1998). The qualitative analysis of chemical bonding exhibits that the U-F bond is more ionic in the relativistic framework (de Jong and Nieuwpoort 1998). The 6s orbital of uranium remains atom-like in the molecule due to relativistic contraction and does not contribute to chemical bonding, while it contributed in nonrelativistic Hartree-Fock theory. 

While relativity stabilizes UF6, stabilization need not always occur. Recently, it has been found that UC 6 is no local minimum within Dirac-Fock theory, while it is stable in quasirelativistic single- and multi-reference calculations (Pyykko etal. 2000). Only four-component multi-reference calculations will give the final answer to die stability of this molecule, in which uranium is in the extraordinary formal oxidation state +XII. 

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B3.1.1.3 WHAT IS LEARNED FROM AN ELECTRONIC STRUCTURE CALCULATION ... 

F) EFFICIENT AND WIDELY DISTRIBUTED COMPUTER PROGRAMS EXIST FOR CARRYING OUT ELECTRONIC STRUCTURE CALCULATIONS... 

This tool, which they call pseudospectralmethods, promises to reduce the CPU, memory and disk storage requirements for many electronic structure calculations, thus pemiitting their application to much larger molecular systems. In addition to ongoing developments in the underlying theory and computer... 

Becke A D 1983 Numerical Hartree-Fock-Slater calculations on diatomic molecules J. Chem. Phys. 76 6037 5 Case D A 1982 Electronic structure calculation using the Xa method Ann. 

Roos B O 1987 The complete active space self-consistent field method and its applications in electronic structure calculations Adv. Chem. Phys. 69 399-445... 

The general potential LAPW teclmiques are generally acknowledged to represent the state of the art with respect to accuracy in condensed matter electronic-structure calculations (see, for example, [62, 73]). These methods can provide the best possible answer within DFT with regard to energies and wavefiinctions. 

Gain G 2000 Large-scale electronic structure calculations using linear scaling methods Status Solidi B 217 231... 

Williams A R, Feibelman P J and Lang N D 1982 Green s-function methods for electronic-structure calculations Phys. Rev. B 26 5433... 

The conceptually simplest approach to solve for the -matrix elements is to require the wavefimction to have the fonn of equation (B3.4.4). supplemented by a bound function which vanishes in the asymptote [32, 33, 34 and 35] This approach is analogous to the fiill configuration-mteraction (Cl) expansion in electronic structure calculations, except that now one is expanding the nuclear wavefimction. While successfiti for intennediate size problems, the resulting matrices are not very sparse because of the use of multiple coordinate systems, so that this type of method is prohibitively expensive for diatom-diatom reactions at high energies. 

M. Peric, B, Engels, and S. D. Peyerimhoff, Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, S. R. Langhoff, ed., Kluwer, Dordrecht, 1995, p. 261. 

Once the least-squares fits to Slater functions with orbital exponents e = 1.0 are available, fits to Slater function s with oth er orbital expon cn ts can be obtained by siin ply m ii Itiplyin g th e cc s in th e above three equations by It remains to be determined what Slater orbital exponents to use in electronic structure calculation s. The two possibilities may be to use the "best atom" exponents (e = 1. f) for II. for exam pie) or to opiim i/e exponents in each calculation. The "best atom expon en ts m igh t be a rather poor ch oicc for mo lecular en viron men ts, and optirn i/.at ion of non linear exponents is not practical for large molecules, where the dimension of the space to be searched is very large.. 4 com prom isc is to use a set of standard exponents where the average values of expon en ts are optirn i/ed for a set of sin all rn olecules, fh e recom -mended STO-3G exponents are... 

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the... 

The Seetion on More Quantitive Aspects of Electronic Structure Calculations introduees many of the eomputational ehemistry methods that are used to quantitatively evaluate moleeular orbital and eonfiguration mixing amplitudes. The Hartree-Foek self-eonsistent field (SCF), eonfiguration interaetion (Cl), multieonfigurational SCF (MCSCF), many-body and Moller-Plesset perturbation theories. 

Semiempirical Methods of Electronic Structure Calculation G. A. Segal, Ed., Plenum, New York (1977). 

W. J. Hehre, Practical Strategies for Electronic Structure Calculations Wavefunction, Ii-vine (1995). 

POLYRATE can be used for computing reaction rates from either the output of electronic structure calculations or using an analytic potential energy surface. If an analytic potential energy surface is used, the user must create subroutines to evaluate the potential energy and its derivatives then relink the program. POLYRATE can be used for unimolecular gas-phase reactions, bimolecular gas-phase reactions, or the reaction of a gas-phase molecule or adsorbed molecule on a solid surface. 

In principle, we could find the minimum-energy crystal lattice from electronic structure calculations, determine the appropriate A-body interaction potential in the presence of lattice defects, and use molecular dynamics methods to calculate ab initio dynamic macroscale material properties. Some of the problems associated with this approach are considered by Wallace [1]. Because of these problems it is useful to establish a bridge between the micro-... 

Computer simulations of electron transfer proteins often entail a variety of calculation techniques electronic structure calculations, molecular mechanics, and electrostatic calculations. In this section, general considerations for calculations of metalloproteins are outlined in subsequent sections, details for studying specific redox properties are given. Quantum chemistry electronic structure calculations of the redox site are important in the calculation of the energetics of the redox site and in obtaining parameters and are discussed in Sections III.A and III.B. Both molecular mechanics and electrostatic calculations of the protein are important in understanding the outer shell energetics and are discussed in Section III.C, with a focus on molecular mechanics. 

J Li, L Noodleman, DA Case. Electronic structure calculations Density functional methods with applications to transition metal complexes. In EIS Lever, ABP Lever, eds. Inorganic Electronic Structure and Spectroscopy, Vol. 1. Methodology. New York Wiley, 1999, pp 661-724. 

In the RISM-SCF theory, the statistical solvent distribution around the solute is determined by the electronic structure of the solute, whereas the electronic strucmre of the solute is influenced by the surrounding solvent distribution. Therefore, the ab initio MO calculation and the RISM equation must be solved in a self-consistent manner. It is noted that SCF (self-consistent field) applies not only to the electronic structure calculation but to the whole system, e.g., a self-consistent treatment of electronic structure and solvent distribution. The MO part of the method can be readily extended to the more sophisticated levels beyond Hartree-Fock (HF), such as configuration interaction (Cl) and coupled cluster (CC). 

Now, let us return to our discussion of carrying out an electronic structure calculation for a nanotube using helical symmetry. The one-electron wavefunc-tions can be constructed from a linear combination of Bloch functions linear combination of nuclear-centered functions Xj(r),... 

In this chapter, we will consider the other half of a model chemistry definition the theoretical method used to model the molecular system. This chapter will serve as an introductory survey of the major classes of electronic structure calculations. The examples and exercises will compare the strengths and weaknesses of various specific methods in more detail. The final section of the chapter considers the CPU, memory and disk resource requirements of the various methods. 

Experimental research chemists with little or no experience with computational chemistry may use this work as an introduction to electronic structure calculations. They will discover how electronic structure theory can be used as an adjunct to their experimental research to provide new insights into chemical problems. 

Part 1, Essential Concepts Techniques, introduces computational chemistry and the principal sorts of predictions which can be made using electronic structure theory. It presents both the underlying theoretical and philosophical approach to electronic structure calculations taken by this book and the fundamental procedures and techniques for performing them. 

Part 3, Applications, discusses electronic structure calculations in the context of real-life research situations, focusing on how it can be used to illuminate a variety of chemical problems. 

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