Molecules finite-difference calculations

Thermochemistry. Chen et al.168 combined the Kohn-Sham formalism with finite difference calculations of the reaction field potential. The effect of mobile ions into on the reaction field potential Poisson-Boltzman equation. The authors used the DFT(B88/P86)/SCRF method to study solvation energies, dipole moments of solvated molecules, and absolute pKa values for a variety of small organic molecules. The list of molecules studied with this approach was subsequently extended182. A simplified version, where the reaction field was calculated only at the end of the SCF cycle, was applied to study redox potentials of several iron-sulphur clusters181. 

The example of a CO molecule has shown that the displacements used in finite-difference calculations for vibrational frequencies should be not too small and not too large. In practice, it is a good idea to choose displacements that result in energy differences on the order of 0.01-0.10 eV since these energy differences can be calculated accurately without requiring extraordinary... 

Finite-Difference Calculations for Atoms and Diatomic Molecules in Strong Magnetic and Static Electric Fields... 

Molecular dynamics consists of the brute-force solution of Newton s equations of motion. It is necessary to encode in the program the potential energy and force law of interaction between molecules the equations of motion are solved numerically, by finite difference techniques. The system evolution corresponds closely to what happens in real life and allows us to calculate dynamical properties, as well as thennodynamic and structural fiinctions. For a range of molecular models, packaged routines are available, either connnercially or tlirough the academic conmuinity. 

The HF results generated for representative polyatomic molecules have used the /V-derivatives estimated by finite differences, while the -derivatives have been calculated analytically, by standard methods of quantum chemistry. We have examined the effects of the electronic and nuclear relaxations on specific charge sensitivities used in the theory of chemical reactivity, e.g., the hardness, softness, and Fukui function descriptors. New concepts of the GFFs and related softnesses, which include the effects of molecular electronic and/or nuclear relaxations, have also been introduced. 

Until recently, only estimates of the Hartree-Fock limit were available for molecular systems. Now, finite difference [16-24] and finite element [25-28] calculations can yield Hartree-Fock energies for diatomic molecules to at least the 1 ghartree level of accuracy and, furthermore, the ubiquitous finite basis set approach can be developed so as to approach this level of accuracy [29,30] whilst also supporting a representation of the whole one-electron spectrum which is an essential ingredient of subsequent correlation treatments. 

To calculate the vibrational frequency of CO using DFT, we first have to find the bond length that minimizes the molecule s energy. The only other piece of information we need to calculate is a = (d2E/db2)h hlj. Unfortunately, plane-wave DFT calculations do not routinely evaluate an analytical expression for the second derivatives of the energy with respect to atomic positions. However, we can obtain a good estimate of the second derivative using a finite-difference approximation ... 

As an example, we can use DFT calculations to evaluate the frequency of the stretching mode for a gas-phase CO molecule using the finite-difference approximation described above. The result from applying Eq. (5.3) for various values of the finite-difference displacement, 8b, are listed and plotted in Table 5.1 and Fig. 5.1. For a range of displacements, say from 8b 0.005 — 0.04 A, the... 

Because the Hessian matrix is calculated in practice using finite-difference approximations, the eigenvalues corresponding to the translational and rotational modes we have just described are not exactly zero when calculated with DFT. The normal modes for a CO molecule with a finite difference displacements of 8b 0.04 A are listed in Table 5.2. This table lists the calculated... 

Btiilding on atomic studies using even-tempered basis sets, universal basis sets and systematic sequences of even-tempered basis sets, recent work has shown that molecular basis sets can be systematically developed until the error associated with basis set truncation is less that some required tolerance. The approach has been applied first to diatomic molecules within the Hartree-Fock formalism[12] [13] [14] [15] [16] [17] where finite difference[18] [19] [20] [21] and finite element[22] [23] [24] [25] calculations provide benchmarks against which the results of finite basis set studies can be measured and then to polyatomic molecules and in calculations which take account of electron correlation effects by means of second order perturbation theory. The basis sets employed in these calculations are even-tempered and distributed, that is they contain functions centred not only on the atomic nuclei but also on the midpoints of the line segments between these nuclei and at other points. Functions centred on the bond centres were found to be very effective in approaching the Hartree-Fock limit but somewhat less effective in recovering correlation effects. 

The applicability of the discussed two-step algorithms for calculation of wavefunctions of molecules with heavy atoms is a consequence of the fact that the valence and core electrons may be considered as two subsystems, interaction between which is described mainly by some integrated properties of these subsystems. The methods for consequent calculation of the valence and core parts of electronic structure of molecules give us a way to combine the relative simplicity and accessibility both of molecular RFCP calculations in gaussian basis set, and of relativistic finite-difference one-center calculations inside a sphere with the atomic core radius. 

The utility of Eq. (9.49) depends on the ease with which the Hessian matrix may be constructed. Methods that allow for the analytic calculation of second derivatives are obviously the most efficient, but if analytic first derivatives are available, it may still be worth the time required to determine the second derivatives from finite differences in the first derivatives (where such a calculation requires that the first derivatives be evaluated at a number of perturbed geometries at least equal to the number of independent degrees of freedom for tlie molecule). If analytic first derivatives are not available, it is rarely practical to attempt to construct the Hessian matrix. 

In any finite field calculation, the choice of F is important because of the conflicting requirements of a small Fto make truncation valid and a large Fto minimize the rounding errors in the differences. The optimum choice clearly depends on the relative values of K0), a, / , etc. for the molecule concerned. 

---