Analytic Derivatives of the Energy

For variationally determined wavefunctions the second term can be shown to vanish, giving the Hellmann-Feynman theorem, 

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The review is organized as follow. In section II we summarize the general basis of the QM continuum solvation models. In section III we present the formal aspects of the PCM, including the theory of the analytical derivatives of the energy. In section IV we present an overview of the PCM approach to the calculation of the properties of molecular solutes. 

It is important to note that the computational schemes that are based on empirical corrections and extrapolation techniques are biased towards the energy of the /i-electron system. They do not improve the -electron wave function and cannot be used if the wave function itself or its expectation values are required. Moreover, it is not necessarily obvious how to compute analytical derivatives of the energy. 

Clearly the whole approach is strictly analogous to the iterative solution of the Roothaan equations, and the approximation itself will have a similar status. The coefficients E, E ,... provide analytic derivatives of the energy with respect to the applied perturbations and the method of obtaining them is much superior to the use of numerical differentiation based on repeated solution of the Roothaan equations for a series of finite values of the parameters (the so-called finite perturbation method of Pople et al., 1976). It is a straightforward matter to extend the treatment to higher orders though the results (Dodds et al. 1977a) become more cumbersome. 

There are many important applications. Thus, by adding field-dependent factors to individual AOs, to obtain field-variant orbitals, it is possible to describe efficiently local polarization effects, in the case of electric fields (see e.g. Sadlej, 1977), and local currents in the case of magnetic fields (see e.g. Jaszunski and Sadlej, 1976, 1977 Jaszunski, 1976) and in the calculation of analytic derivatives of the energy with respect to nuclear displacements (see e.g. Gerratt and Mills, 1968 Pulay, 1977) it is possible to obtain accurate force constants and energy surfaces. 

Indeed, the A2 amplitudes are not required to determine the CCD energy. They arise in analytical derivatives of the energy as the so-called Z-vector (75). 

A drawback of the SCRF method is its use of a spherical cavity molecules are rarely exac spherical in shape. However, a spherical representation can be a reasonable first apprc mation to the shape of many molecules. It is also possible to use an ellipsoidal cavity t may be a more appropriate shape for some molecules. For both the spherical and ellipsoi cavities analytical expressions for the first and second derivatives of the energy can derived, so enabling geometry optimisations to be performed efficiently. For these cavil it is necessary to define their size. In the case of a spherical cavity a value for the rad can be calculated from the molecular volume ... 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

Molecular frequencies depend on the second derivative of the energy with respect to the nuclear positions. Analytic second derivatives are available for the Hartree-Fock (HF keyword). Density Functional Theory (primarily the B3LYP keyword in this book), second-order Moller-Plesset (MP2 keyword) and CASSCF (CASSCF keyword) theoretical procedures. Numeric second derivatives—which are much more time consuming—are available for other methods. 

Numerical optimizations are available for methods lacking analytic gradients (first derivatives of the energy), but they are much, much slower. Similarly, frequencies may be computed numerically for methods without analytic second derivatives. 

Derivative techniques consider the energy in the presence of the perturbation, perform an analytical differentiation of the energy n times to derive a formula for the nth-order property, and let the perturbation strength go to zero. 

The tensor of the static first hyperpolarizabilities P is defined as the third derivative of the energy with respect to the electric field components and hence involves one additional field differentiation compared to polarizabilities. Implementations employing analytic derivatives in the Kohn-Sham framework have been described by Colwell et al., 1993, and Lee and Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic derivatives are available, some finite field approximation is used. In these cases the P tensor is preferably computed by numerically differentiating the analytically obtained polarizabilities. In this way only one non-analytical step, susceptible to numerical noise, is involved. Just as for polarizabilities, the individual tensor components are not regularly reported, but rather... 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

The difference between this result and the thermodynamic perturbation result is that in the former (eq. (11.28)) the average is taken of (finite) differences in energy while eq. (11.31) averages over a differentiated energy function often the required derivative of the energy with respect to the coupling parameter can be obtained analytically and the averaging involved here is no more complicated than with eq. (11.28). 

The calculations described in this chapter were performed using one of the Gaussian programs from Pople s group and the CADPAC program of Handy. The current versions of these programs calculate the second derivative of the energy analytically. 

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 ... 

The selection rules for the QM harmonic oscillator pennit transitions only for An = 1 (see Section 14.5). As Eq. (9.47) indicates diat the energy separation between any two adjacent levels is always hm, the predicted frequency for die = 0 to n = 1 absorption (or indeed any allowed absorption) is simply v = o). So, in order to predict die stretching frequency within the harmonic oscillator equation, all diat is needed is the second derivative of the energy with respect to bond stretching computed at die equilibrium geometry, i.e., k. The importance of k has led to considerable effort to derive analytical expressions for second derivatives, and they are now available for HF, MP2, DFT, QCISD, CCSD, MCSCF and select other levels of theory, although they can be quite expensive at some of the more highly correlated levels of theoiy. 

With the development of analytical energy derivative methods135 l67, the calculation of vibrational frequencies (second derivatives of the energy with regard to atomic coordinates) and infrared absorption intensities (derivatives of the energy with regard to components of electronic field and atomic coordinates, i.e. dipole moment derivatives) both at the HF and correlation corrected levels has become routine168. There are six (two a " + four e)... 

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