Calculation of Expectation Values

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

Kello, V. and Sadlej, A.J. (1998) Picture change and calculations of expectation values in approximate relativistic theories. International Journal of Quantum Chemistry, 68, 159—174. 

Bemardi and Boys49 have examined the problem of the accuracy of the energy and other variables in this method, and give explicit formulae for improving the calculations. The original formulation of the method to cover the calculation of expectation values was given by Handy and Epstein in 1970.50 Armour 51 has examined the method of moments and the transcorrelated wavefunction method (which is a particular form of the method of moments) in some detail. Several expectation values were evaluated in the course of applications of the former method to H2, and in general fairly accurate results were obtained, but numerical problems can occur, and further study is needed. 

In the MCSCF case the undifferentiated Fock matrix is symmetric since the orbital optimization ensures that ffj = 2(FfJ — FfJ) = 0. The Fock matrix also appears in the calculation of expectation values of one-index transformed Hamiltonians (see Appendix F). 

When used with methods that include electronic correlation, the supermolecular approach accounts for all interaction terms, charge transfer included. The value of the various contributions, however, is not separately known and decomposition procedures [25-28] have been proposed to this end, the most widely applied being due to Morokuma [25]. Generally speaking, these procedures rely on the calculation of expectation values of the energy in states described by eigenfunctions that are products of eigenfunctions of the separated species. 

For example, the wavefimction (5.15) is not changed if we replace (pu by Xi = V ls + that is, if we replace any row of the determinant by a linear combination of this row with other rows. Such mathematical alterations inside the determinant do not correspond to any physical change, because the wavefimction ip remains unchanged. Therefore, atomic orbitals truly only have a mathematical meaning in so far as they are the basis of the wave-function Ip expressed as a Slater determinant (however, see Section 5.4). It is this wavefimction which has physical meaning through ip ip in the calculation of expectation values ... 

Given that a probability distribution can be characterized by a mean and variance, this section will discuss the nature of these parameters. The following section will discuss how to estimate these parameters from a sample. For virtually any probability density function, the average or mean and variance can be determined by the calculation of expected values. The expected value of, v for the distribution/(.r) is given by ... 

The calculation of expectation values of operators over the wavefunction, expanded in terms of these determinants, involves the expansion of each determinant in terms of the N expansion terms followed by the spatial coordinate and spin integrations. This procedure is simplified when the spatial orbitals are chosen to be orthonormal. This results in the set of Slater Condon rules for the evaluation of one- and two-electron operators. A particularly compact representation of the algebra associated with the manipulation of determinantal expansions is the method of second quantization or the occupation number representation . This is discussed in detail in several textbooks and review articles - - , to which the reader is referred for more detail. An especially entertaining presentation of second quantization is given by Mattuck . The usefulness of this approach is that it allows quite general algebraic manipulations to be performed on operator expressions. These formal manipulations are more cumbersome to perform in the wavefunction approach. It should be stressed, however, that these approaches are equivalent in content, if not in style, and lead to identical results and computational procedures. 

The computation of the coupling coefficients is of utmost importance in the calculation of expectation values and of certain matrix-vector products. There are basically two different approaches that may be used in the evaluation of these terms. These terms may be computed once and stored as a separate file, which is read repeatedly when required, or they may be repeatedly computed and used as they are required. The first scheme has the advantage that any overhead associated with the repeated construction of these coupling coefficients is minimized. The second method has the advantage that no potentially large external files are required as in the first method. Both approaches have been used in MCSF calculations and the optimal approach is computer-dependent. 

These density functions can be used in the calculation of expectation values rather than using the more commonly known formula based on wave functions. For the different observables, one has a formula [51] ... 

For some applications, e.g. the calculation of expectation values of non-multiplicative operators, like kinetic energy, one needs the density matrices q i, ri) and 7i(ri, 2, 7s) rather than the density functions. 

The solution of Eq. (11.60) is achieved by standard methods once the matrix elements are obtained. Their evaluation involves the calculation of expectation values, which can be found when the solution of Eq. (11.60) is available, and we are, thus, faced with a self-consistency requirement, which is similar to but more complex than the corresponding challenge in the Hartree-Fock approximation. In the next section, the calculation of the matrix elements is addressed. 

The two-component methods, though much simpler than the approaches based on the 4-spinor representation, bring about some new problems in calculations of expectation values of other than energy operators. The unitary transformation U on the Dirac Hamiltonian ho (Eq.4.23 is accompanied by a corresponding reduction of the wave function to the two-component form (Eq.4.26). The expectation value of any physical observable 0 in the Dirac theory is defined as ... 

However, in two-component quantum chemical calculations of expectation values the unitary transformation of the operator O is often not taken into account. Instead, the quantity... 

We will describe the calculation of expectation values like the energy, dipole moment, etc., and other quantities of interest like population analyses shortly (Subsection 3.4.7) but let us first consider some of the practical questions involved in each of the twelve steps. 

Finite-difference methods operating on a grid consisting of equidistant points ( Xi, Xi = ih + Xq) are known to be one of the most accurate techniques available [496]. Additionally, on an equidistant grid all discretized operators appear in a simple form. The uniform step size h allows us to use the Richardson extrapolation method [494,497] for the control of the numerical truncation error. Many methods are available for the discretization of differential equations on equidistant grids and for the integration (quadrature) of functions needed for the calculation of expectation values. 

In the following sections, we introduce the basic principles of the correct calculation of expectation values for a single electron moving in external fields. 

V. KeUo, A. J. Sadlq. Picture Change and Calculations of Expectation Values in Approximate Relativistic Theories. Int. ]. Qmntum Chem., 68 (1998) 159-174. 

In Section 12.2 it will be discussed that this approach for the calculation of expectation values is called the unrelaxed method, because the conditions for the molecular orbital coefficients were not included as additional constraints in the coupled cluster Lagrangian given in Eq. (9.95) or Eq. (9.98). A coupled cluster Lagrangian including orbital relaxation takes the following form... 

In the calculation of expectation values it is convenient to introduce the so called bra 4> and ket 4>) notation with the first representing the wavefunction and the second its complex conjugate. In the bra and ket expressions the spatial coordinate r is left deliberately unspecified, so that they can be considered as wavefunctions independent of the representation when the coordinate r is specified, the wavefunctions are considered to be expressed in the position representation . Thus, the expectation value of an operator O in state

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An important application of these rules is the calculation of expectation values of operators in the basis of states with definite J and J. Since these states are obtained by combinations of states with spin S and orbital angular momentum L, we denote them as JLSJ ), which form a complete set of (2/ + 1) states for each value of J, and a complete set for all possible states resulting from the addition of L and S when all allowed values of J are included ... 

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