Wave function second-order derivatives

We have established that, for a fully variational wave function, we may calculate the first-order properties from the zero-order response of the wave function (i.e., from the unperturbed wave function) and the second-order properties from the first-order response of the wave function. In general, the 2n -f 1 rule is obeyed For fully variational wave functions, the derivatives (i.e., responses) of the wave function to order n determine the derivatives of the energy to order 2n+ 1. This means, for instance, that we may calculate the energy to third order with a knowledge of the wave function to first order, but that the calculation of the energy to fourth order requires a knowledge of the wave-function response to second order. 

If A is a slowly varying function of z, we can neglect the second-order derivative of it with respect to z, thus obtaining the Fock-Leontovich parabolic wave equation... 

Although a calculation of the wave function response can be avoided for the first derivative, it is necessary for second (and higher) derivatives. Eq. (10.29) gives directly an equation for determining the (first-order) response, which is structurally the same as eq. (10.36). For an HF wave function, an equation of the change in the MO coefficients may also be formulated from the Hartree-Fock equation, eq. (3.50). 

As these expressions correspond to the CC energy derivative, they must give size-extensive results. However, the price we pay is that the energy of a given order requires wave function contributions of the same order. Furthermore, these non linear terms are difficult to evaluate. The quadartic in term in second-order, requires comparable difficulty to the quadratic terms in a CCSD calculation... 

The product of a function and its complex conjugate is always real and is positive everywhere. Accordingly, the wave function itself may be a real or a complex function. At any point x or at any time t, the wave function may be positive or negative. In order that F(x, t)p represents a unique probability density for every point in space and at all times, the wave function must be continuous, single-valued, and finite. Since F(x, /) satisfies a differential equation that is second-order in x, its first derivative is also continuous. The wave function may be multiplied by a phase factor e , where a is real, without changing its physical significance since... 

We shall in this chapter discuss the methods employed for the optimization of the variational parameters of the MCSCF wave function. Many different methods have been used for this optimization. They are usually divided into two different classes, depending on the rate of convergence first or second order methods. First order methods are based solely on the calculation of the energy and its first derivative (in one form or another) with respect to the variational parameters. Second order methods are based upon an expansion of the energy to second order (first and second derivatives). Third or even higher order methods can be obtained by including more terms in the expansion, but they have been of rather small practical importance. 

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 ... 

These wave functions, must be well-behaved, that is, they (and their derivatives with respect to the space coordinates) must be continuous, finite, and single valued. The functions are solutions to a second-order differential equation called the Schrodinger equation (see below). 

Over the decade 1995-2005, ab initio quantum chemistry has become an important tool in studying imidazole derivatives. Two highly productive approaches are often utilized for the calculations the wave function-based methods (e.g., Hartree-Fock theory and second-order Moller-Plesset perturbation theory (MP2)) and the density functional theory (DFT) based methods (e.g., gradient-corrected (BLYP) and hybrid (B3LYP) methods). 

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