Hessian expressions

The Cl gradient expression was derived and implemented by Krishnan et al. (1980) and Brooks et al. (1980). The generalization to MRCI is due to Osamura et al. (1981, 1982a,b). The Hessian expression was derived by Jorgensen and Simons(1983) and implemented by Fox et al. (1983). Recently, a more efficient implementation has been reported by Lee et al. (1986). MRCI derivative expressions up to fourth order have been derived by Simons et al. (1984). The introduction of the Handy-Schaefer technique (Handy and Schaefer, 1984) greatly improved the efficiency of Cl derivative calculations. The calculation of Cl derivatives within the Fock-operator formalism has recently been reviewed by Osamura et al. (1987). 

The CC molecular gradient and Hessian expressions were derived by Jorgensen and Simons (1983). Using the Handy-Schaefer technique, Adamowicz et al. (1984) and Bartlett (1986) simplified the expressions for the gradient. The only implementations are the CC molecular gradients reported by Fitzgerald et al. (1985) and by Lee et al. (1987). 

The MP2 gradient expression was derived and implemented by Pople et al. (1979). Second derivative expressions were given by J0rgensen and Simons (1983). Handy et al. (1985, 1986) and Harrison et al. (1986) simplified the gradient and Hessian expressions using the Handy- Schaefer technique and reported implementations of these expressions. 

Although the Lanczos is a fast efficient algorithm, it does not necessarily give savings in memory. To save memory a number of techniques divide the molecule into smaller parts that correspond to subspaces within which the Hessian can be expressed as a matrix of much lower order. These smaller matrices are then diagonalized. The methods described below show how one then proceeds to achieve good approximations to the true low frequency modes by combining results from subspaces of lower dimension. 

Many gradient methods approximate the energy surface at step by a quadratic expression in terms of the coordinate vector the total energy the gradient and the Hessian... 

The above formula is obtained by differentiating the quadratic approximation of S(k) with respect to each of the components of k and equating the resulting expression to zero (Edgar and Himmelblau, 1988 Gill et al. 1981 Scales, 1985). It should be noted that in practice there is no need to obtain the inverse of the Hessian matrix because it is better to solve the following linear system of equations (Peressini et al. 1988)... 

The four blocks of V can be alternatively expressed in terms of the principal geometric derivatives defining the generalized Hessian of Equation 30.8. This can be accomplished first by expressing AQ as function of AN and AF, using the second Equation 30.9, and then by inserting the result into the first Equation 30.9 ... 

Inserting the first- and second-order wavefiinction variations into either of the expressions Eq. (6) or (12) yields the complete expressions for the gradient and Hessian. In doing this, it becomes clear that the computationally most expensive parts are related to the terms containing first-order variations in both bra and ket, e.g. 

There are a few points with respect to this procedure that merit discussion. First, there is the Hessian matrix. With elements, where n is the number of coordinates in the molecular geometry vector, it can grow somewhat expensive to construct this matrix at every step even for functions, like those used in most force fields, that have fairly simple analytical expressions for their second derivatives. Moreover, the matrix must be inverted at every step, and matrix inversion formally scales as where n is the dimensionality of the matrix. Thus, for purposes of efficiency (or in cases where analytic second derivatives are simply not available) approximate Hessian matrices are often used in the optimization process - after aU, the truncation of the Taylor expansion renders the Newton-Raphson method intrinsically approximate. As an optimization progresses, second derivatives can be estimated reasonably well from finite differences in the analytic first derivatives over the last few steps. For the first step, however, this is not an option, and one typically either accepts the cost of computing an initial Hessian analytically for the level of theory in use, or one employs a Hessian obtained at a less expensive level of theory, when such levels are available (which is typically not the case for force fields). To speed up slowly convergent optimizations, it is often helpful to compute an analytic Hessian every few steps and replace the approximate one in use up to that point. For really tricky cases (e.g., where the PES is fairly flat in many directions) one is occasionally forced to compute an analytic Hessian for every step. 

If we had estimates in hand, the simplest way to estimate the expected values of the Hessian would be to evaluate the expressions above at the maximum likelihood estimates, then compute the negative inverse. First, since the expected value of <51nL/3a is zero, it follows that E[x/5] = 1/a. Now,... 

In this section we shall compute the energy gradient and the Hessian matrix corresponding to the energy expression (3 25). We introduce the variation of the Cl coefficients by operating on the MCSCF state I0> with the unitary... 

The order of the operators in (4 5) is not arbitrary, since they do not commute. The reverse order, however, leads to more complicated expressions for the Hessian matrix, and since the final result is independent of the order, we make the more simple choice given in (4 5). The energy corresponding to the varied state (4 5) will be a function of the parameters in die unitary operators, and we can calculate the first and second derivatives of this function... 

We now have the expressions for the gradient and the Hessian matrix. The corresponding Newton-Raphson equations can then be written down in matrix form as ... 

Now choose A0 to be the diagonal part of the Hessian matrix and A the remaining non-diagonal part. We then obtain for the Cl part of the update vector y the following expression ... 

We shall in this section derive the explicit expressions for the elements of the gradient vector and the Hessian matrix. The derivation is a good exercise in handling the algebra of the excitation operators fey and the reader is suggested to carry out the detailed calculations, where they have been left out in the present exposition. 

Derive the detailed expression for the orbital Hessian for the special case of a closed shell single determinant wave function. Compare with equation (4 53) to check the result. The equation can be used to construct a second order optimization scheme in Hartree-Fock theory. What are the advantages and disadvantages of such a scheme compared to the conventional first order methods ... 

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