Calculations structures

Molecular modeling is extensively covered in other articles, and since the interactions between molecules in crystals are the same as in other aggregation states of matter, no really new principles are involved. Therefore, the following discussion is limited to subjects that are direcdy related to the symmetry of crystals. 

In the simulation of liquids it is usual to consider a computational box with periodic boundary conditions. For the calculation of static crystal energies and forces, this box is naturally a unit cell, but more advantage of symmetry can be taken by limiting the attention to the asymmetric unit only. However, to simulate dynamic properties it may be necessary to use a computational box that comprises a number of unit cells. Calculation of free energies is, as always, difficult but there are a few approaches that are not applicable to the liquid state. A possible complication, not encountered in liquids, is that the unit cell may have a net dipole moment. In that case one must note that calculations with a simple cut-off scheme do not produce the same result as Ewald summations. 

The anisotropy of the crystal necessitates a reconsideration of the usual concept of pressure, and complicates the calculation of volume changes that occur under the influence of an external pressure. One is rewarded by obtaining simulation results that can be compared with accurately known cell parameters from crystallography. In many cases the internal crystal structure is also known with great accuracy, and a comparison of simulated and measured structural details is a very important criterion for the adequacy of a force field. Recently an exciting new requirement has been added the force field should enable us to predict why a certain polymorph is observed rather than any other one out of many hypothetical possibilities. This fascinating problem is still to a large extent unsolved. 

In the calculation of Coulomb and dispersion terms for a crystal we consider one representative unit cell with N atoms and volume V. Let rt be the vector from the cell origin to an atom k in the cell, and ri /i- the distance from that atom to an atom k in another cell which is related to the first cell by a translation / (including, of course, the case i = 1 no translation). Then most contributions to the nonbonded energy can be written as 

In practice, of course, the number of terms is limited by considering only contributions within a certain interatomic distance, the cut-off radius. For = 6 we have the dispersion energy which converges fairly well yet a cut-off distance of 10 A may be insufficient for accurate work. Although each individual term beyond that distance is small, there are many contributions and they are all attractive. For n = 1 we have the 

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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. 

Terakura K, Qguchi T, Williams A R and Kubler J 1984 Band theory of insulating transition-metal monoxides Band-structure calculations Phys. Rev. B 30 4734... 

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... 

Cortona P 1991 Self-consistently determined properties of solids without band structure calculations Phys. Rev. B 44 8454... 

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. 

Guntert P 1998 Structure calculation of biological macromolecules from nmr data 1998 Q. Rev. Biophys. 31 145-237... 

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... 

Schematic representation of some of the lower frequencies in the ion-dipole complex for the Cl + MeCl m and the imaginary frequency of the transition structure, calculated using a 6-31G basis set. 

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. 

The primary reason for interest in extended Huckel today is because the method is general enough to use for all the elements in the periodic table. This is not an extremely accurate or sophisticated method however, it is still used for inorganic modeling due to the scarcity of full periodic table methods with reasonable CPU time requirements. Another current use is for computing band structures, which are extremely computation-intensive calculations. Because of this, extended Huckel is often the method of choice for band structure calculations. It is also a very convenient way to view orbital symmetry. It is known to be fairly poor at predicting molecular geometries. 

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). 

The simplest approximation to the complete problem is one based only on the electron density, called a local density approximation (LDA). For high-spin systems, this is called the local spin density approximation (LSDA). LDA calculations have been widely used for band structure calculations. Their performance is less impressive for molecular calculations, where both qualitative and quantitative errors are encountered. For example, bonds tend to be too short and too strong. In recent years, LDA, LSDA, and VWN (the Vosko, Wilks, and Nusair functional) have become synonymous in the literature. 

Molecular mechanics methods are not generally applicable to structures very far from equilibrium, such as transition structures. Calculations that use algebraic expressions to describe the reaction path and transition structure are usually semiclassical algorithms. These calculations use an energy expression fitted to an ah initio potential energy surface for that exact reaction, rather than using the same parameters for every molecule. Semiclassical calculations are discussed further in Chapter 19. 

Since transition-structure calculations are so sensitive to the starting geometry, a number of automated techniques for finding reasonable starting geometries have been proposed. One very useful technique is to start from the reactant and product structures. 

Band structure calculations have been done for very complicated systems however, most of software is not yet automated enough or sufficiently fast that anyone performs band structures casually. Setting up the input for a band structure calculation can be more complex than for most molecular programs. The molecular geometry is usually input in fractional coordinates. The unit cell lattice vectors and crystallographic angles must also be provided. It may be nee-... 

Any orbital-based scheme can be used for crystal-structure calculations. The trend is toward more accurate methods. Some APW and Green s function methods use empirical parameters, thus edging them toward a semiempirical classification. In order of preference, the commonly used methods are ... 

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