Wave function first-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. 

In such cases the expression from fii st-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, MP or CC), the Hellmann-Feynman theorem does not hold, and a first-order property calculated as an expectation value will not be identical to that obtained as an energy derivative. Since the Hellmann-Feynman theorem holds for an exact wave function, the difference between the two values becomes smaller as the quality of an approximate wave function increases however, for practical applications the difference is not negligible. It has been argued that the derivative technique resembles the physical experiment more, and consequently formula (10.21) should be preferred over (10.18). 

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). 

If the perturbation is a homogeneous electric field F, the perturbation operator P i (eq. (10.17)) is the position vector r and P2 is zero. As.suming that the basis functions are independent of the electric field (as is normally the case), the first-order HF property, the dipole moment, from the derivative formula (10.21) is given as (since an HF wave function obeys the Hellmann-Feynman theorem)... 

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... 

Equation 2.24 can be thought of as having been derived from Equation 2.25 by adding the third term on the left hand side of Equation 2.24 as a perturbation. In first order quantum mechanical perturbation theory (see any introductory quantum text), the perturbation on the ground state of Equation 2.25 is obtained by averaging the perturbation over the ground state wave function of Equation 2.25. The effect of this... 

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 ... 

Relativity becomes important for elements heavier than the first row transition elements. Most methods applicable on molecules are derived from the Dirac equation. The Dirac equation itself is difficult to use, since it involves a description of the wave function as a four component spinor. The Dirac equation can be approximately brought to a two-component form using e.g. the Foldy-Wouthuysen (FW) transformational,12]. Unfortunately the FW transformation, as originally proposed, is both quite complicated and also divergent in the expansion in the momentum (for large momenta), and it can thus only be carried out approximately (usually to low orders). The resulting equations are not variationally stable, and they are used only in first order perturbation theory. 

Fig. 7. (a) Wave pattern generated on a circle with scaled radius kr = 1.82. The pattern is the product of a radial part, J,(kr) [the first-order Bessel function] and an angular part, cos ( >- The dashed nodal line of zero (i.e., steady state) concentration runs along diameter of the circle from < > = 90° to < > = 270°. The dotted circle outlines the circular radius. (b) Wave pattern, JAkr) cos 2dashed lines are crossed nodal lines on two perpendicular diameters, (c) Pattern generated at a scaled radius of 3.8, where the zero in the derivative of JAkr) matches the radial boundary condition. The pattern is JAkr) cos 0c >, which has no angular variation. The nodal line is concentric with the outer radius. 

These equations are known as the response equations since they determine the first derivatives (i.e., the first-order response) of the wave function to the perturbation. 

In such cases the expression from first-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, ME or CC), the Hellmann-Feymnan theorem does not hold, and a first-order property calculated as an... 

An equation for the llO) elements can be obtained from the condition that the Fock matrix is diagonal, and expanding all involved quantities to first-order. and solving the CPHF equations is usually not the bottleneck in these cases. Without the Lagrange technique for non-variational wave functions (Cl, MP and CC), the nth-order CPHF is needed for the nth-derivative. Consider for example the MP2 energy correction, eq. (4.45). ... 

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