Expectation values Hellmann-Feynman theorem

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 the expectation value r )ni cannot be obtained from equation (6.70), it can be evaluated by regarding the azimuthal quantum number I as the parameter in the Hellmann-Feynman theorem (equation (3.71)). Thus, we have... 

By the Hellmann-Feynman theorem, the expectation value < f -dV/dRua I f) is the force on nucleus N in the a direction. The force on each nucleus vanishes for a molecule in its equilibrium nuclear configuration the force also vanishes for an isolated atom. In these cases the virial theorem becomes (T) = -other cases, however, the second term on the right in Eq. (17) is non-vanishing. 

Equation (9.32) is also useful to the extent it suggests die general way in which various spectral properties may be computed. The energy of a system represented by a wave function is computed as the expectation value of the Hamiltonian operator. So, differentiation of the energy with respect to a perturbation is equivalent to differentiation of the expectation value of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized with respect to the coefficients defining die wave function, the Hellmann-Feynman theorem of quantum mechanics allows us to write... 

A useful expression for evaluating expectation values is known as the Hellmann-Feynman theorem. This theorem is based on the observation that the Hamiltonian operator for a system depends on at least one parameter X, which can be considered for mathematical purposes to be a continuous variable. For example, depending on the particular system, this parameter X may be the mass of an electron or a nucleus, the electronic charge, the nuclear charge parameter Z, a constant in the potential energy, a quantum number, or even Planck s constant. The eigenfunctions and eigenvalues of H(X) also depend on this... 

If the zero-order wavefunction does, in fact, satisfy Eqn. (26), then the first term on the right in Eqn. (32) is identically zero. This means that the first derivatives are obtained as an expectation value of the derivative Hamiltonian. This last statement is the Hellmann-Feynman theorem. 

According to the Hellmann-Feynman theorem [ 18,22], when a Hamiltonian depends on a parameter A, the derivative of the energy with respect to A is equal to the expectation value of the derivative of the Hamiltonian with respect to A,... 

The Hellmann-Feynman theorem suggests that molecular properties can also be defined from a perturbation expansion of expectation values. Generally, the energy derivative formalism is to be preferred over the expectation value approach as the former facilitates a straightforward extension to non-variational wave functions through the introduction of the Lagrangian [8,9]... 

The second possibility is to use a gradient code, if this is available for the chosen method. The third method is the simplest, namely to evaluate E2 as an expectation value. This method is equivalent to the other two, if Eq satisfies a stationarity condition, like the Brillouin condition of Hartree-Fock theory. For non-stationary approaches, like MP2 or CC, the methods based on differentiations of the expectation value are more reliable. This is related to the validity or non-validity of the Hellmann-Feynman theorem. 

In contrast to variational wave functions where the first order response El — dE Q)/dQ)Q equals the expectation value of the perturbation operator with the unperturbed wave function (Hellmann-Feynman theorem) a general expression has to be used in combination with non-variational wave functions derived by differentiating all Q-dependent terms in eq. (55) ... 

Care must be taken in using the expressions above for obtaining nonlinear optical properties, because the values obtained may not be the same as those obtained from Eq. [4]. The results will be equivalent only if the Hellmann-Feyn-man theorem is satisfied. For the case of the exact wavefunction or any fully variational approximation, the Hellmann-Feynman theorem equates derivatives of the energy to expectation values of derivatives of the Hamiltonian for a given parameter. If we consider the parameter to be the external electric field, F, then this gives dE/dP = dH/d ) = (p,). For nonvariational methods, such as perturbation theory or coupled cluster methods, additional terms must be considered. 

The first term is the Hellmann-Feynman contribution and is readily calculated as an expectation value of a one-electron operator. The second part can be called the wavefunction derivative term (it has also come to be known as the Pulay term). Typical MO wavefunctions are constructed from basis functions that are centered on the atoms and follow them rigidly. Such wavefunctions do not obey the Hellmann-Feynman theorem and the wavefunction derivative term is not zero. Many molecular properties can also be written as energy derivatives [1-4]. For some properties, it is advantageous to make the wavefunction depend on the electric or magnetic field for these cases the wavefunction derivative term must also be calculated. 

This general result is well known as the "Hellmann-Feynman" theorem when X represents the position x of a nucleus. The force F that the system exerts on the nucleus is the expectation value of minus the gradient of V(x), where V is the potential that acts on the nucleus. This theorem was originally derived by Ehrenfest (1927), and was used in Hellmann s (1937) treatise to establish the forces in a molecule. Feynman (1939) independently derived the result for molecules. We will refer to the result simply as the "force theorem". 

Substitution of 5H into the formal Hellmann-Feynman theorem of Eq. (14.4) gives the variation of the electronic energy as the expectation value ... 

The exact eigenfunctions of the effective PCM Hamiltonian (1.12) obey to a generalized Hellmann-Feynman, theorem according to which the first derivative of the free-energy functional G (1.10) with respect to a perturbation parameter k may be compute as expectation value with the unperturbed wavefunction ... 

Before starting properly with perturbation theory we will first introduce in the next section the Hellmann Feynman theorem, which establishes a deep connection between the energy and molecular properties calculated as expectation values and that does not rely on perturbation theory. 

Alternatively, we can employ the Hellmann-Feynman theorem, Eq. (3.11), and obtain the cartesian components of the electric moments directly as expectation values of the corresponding derivatives of the Hamiltonian... 

The derivation of a component of the magnetic dipole moment as first derivative of the perturbed energy, Elq. (5.18), via perturbation theory or the Hellmann Feynman theorem then leads to the following expectation value... 

In non-variational approaches such as Moller-Plesset perturbation theory or coupled cluster methods the wavefunction is not at all variationally optimized. However, it is possible to choose ( o hi such a way that the Hellmann-Feynman theorem is fulfilled to a certain extent, while the transition expectation value in Eq. (9.88) still gives the energy. 

The transition expectation value Eq , or coupled cluster Lagrangian Lq, is thus stationary with respect to the configuration or determinant coefficients and therefore satisfies partially the Hellmann-Feynman theorem... 

The Hellmann-Feynman theorem (J. Hellmann. Einfiihrung in die Quantenchemie , Deuticke, Leipzig, 1937 R. P, Feynman, Phys. Rev, 1939, 56, 340-343) states that first-oider properties can be evaluated as simple expectation values of the unperturbed wavefunction over the corresponding perturbed operator V. However, it should be noted that the Hellmann-Feynman theorem is not satisfied for truncated CC approaches and that evaluation of the energy derivatives appears to be the preferred choice. Furthermore, in ca.ses where additional wavefunction parameters such as, for example, the supplied basis function, exhibit a perturbation dependence, the Hellmann-Feynman theorem is not valid as long us finite basis sets are used. 

Using the Hellmann-Feynman theorem it can be shown that the negative of the first derivative of the energy with respect to the applied field is given by the expectation value of the dipole operator. This is referred to as the field-dependent dipole moment and can be expressed as ... 

Equation (13.5.8), which represents the generalization of the Hellmann-Feynman theorem to coupled-cluster wave functions, is shown in Exercise 13.1 to give size-extensive first-order properties. For variational wave functions, the Hellmann-Feynman theorem contains the real average value of the operator V (4.2.51) rather than a transition expectation value as in (13.5.8). Likewise, to ensure teal properties, we may in coupled-cluster theory work in terms of the manifestly real expression... 

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