Expectation value of the energy

In practice, one is restricted to energies between and and a range of temperatures T min E T < Tmax where the range of permitted temperatures is determined by calculating the expectation value of the energy at temperature T ... 

Suppose we have a wavefunction P, which for the sake of argument I will assume to be real. The expectation value of the energy is... 

In order to evaluate the expectation value of the energy for an electronic system it is hence sufficient to know the generalized second-order density matrix r(x x 2 x1x2), from which the first-order density matrix may be obtained by using the formula... 

Let us now keep the parameter R fixed and vary rj in the expectation value of the energy... 

In Section II.B, we have used the density matrices to simplify the calculations, but the wave functions W are still the fundamental quantities. Relation II. 11 shows,however, that the expectation value of the energy p)Av depends only on the second-order density matrix, and we can rewrite it in the form22... 

The statistical matrix is then computed via Eq. (7-78). When the expectation value of the energy, Tr (pH), is then calculated in different orders of V, the successive orders of time-dependent perturbation theory emerge. 

In practice, using trial wave functions with one or more variable parameters, the true ground state may be approached by minimizing the expectation value of the energy with respect to these parameters. 

If the given Hamiltonian operates on the product wave function, ft only operates on ip(i), eic., and the contribution of this wave function to the expectation value of the energy becomes... 

Using an anti-symmetrized, rather than a simple product of one-electron orbitals, the expectation value of the energy becomes a sum of one-electron, Coulomb and exchange terms ... 

For a closed system, therefore, the expectation value of the energy is constant in time, which is the energy theorem of quantum mechanics. 

Now expectation values of the energies are calculated from all the four wave functions. The individual contributions to the energy of interaction are obtained as differences ... 

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

If a system is in a state described by a wave function ip xi,yi, z, s. .. x ., Vn, Zn, Sn) that is Hot an eigenfunction of the Hamiltonian, measurements of the energy will give variable results. An expression for the average, or expectation value, of the energy measurements is the integral ... 

Problem 8-12. Verify that if -ip is an eigenfunction of H with eigenvalue , then the expectation value of the energy is equal to . The expression (/, g) is called the inner product of fund g. It has a number of properties analogous to those of the dot product of two vectors. These are illustrated in Table 8.2. 

Calculate the difference between the expectation value of the energy given by the approximate wave function and the known true energy, or... 

To get some idea of the use of trial wave functions and the variation principle, evaluate the expectation value of the energy using the Hydrogen atom Hamiltonian, and normalized Is orbitals with variable Z. That is, evaluate ... 

The expectation value of the energy calculated with is greater than or equal to L. 

The variation theorem has been an extremely powerful tool in quantum chemistry. One important technique made possible by the variation theorem is the expression of a wave function in terms of variables, the values of which are selected by minimizing the expectation value of the energy. 

Let s assume a wave function of the Slater determinant form and find an expression for the expectation value of the energy. We ve written a Slater determinant as a ket vector in shorthand notation, allowing us to make use of Dirac notation for such things as overlap. In this context, recall that... 

Next, we focus on the dynamics after ultrafast IR excitation of the main absorption peaks (cf. Fig. 1). The laser pulse spectra are shown in Fig. 2. Clearly, they are broad enough to excite a coherent superposition, i.e., a vibrational wave packet, in both cases. The dynamics is analyzed using the expectation values of the energy of the H/D reaction coordinates, Exy(t) = Ea a( I (t) a) a I (t)), and of the energies of the uncoupled skeleton modes,... 

Since the functions dy are fixed, the expectation value of the energy E considered as a function of the variational parameters u, is a quadratic fonn ... 

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