Generalized Hellmann-Feynman

This modihed amplitude equation does not correspond to the minimization of the energy functional Eq. (7), and thus the generalized Hellmann-Feynman theorem [49] does not apply. 

Hellmann-Feynman and virial theorems 4.2.1 Generalized Hellmann-Feynman theorem... 

The electrostatic Hellmann-Feynman theorem is a special form of the general Hellmann-Feynman theorem. This form of the theorem can be expressed in terms of electronic density, and no explicit form of the electronic wave function is needed. The electrostatic Hellmann-Feynman theorem is of special significance in view of new developments in the construction of macromolecular electron densities and density matrices without using wave functions. ... 

In a truncated coupled cluster approach, tlie two vectors are not connected by the adjoint operation but without truncations a representation of the exact state situation is retrieved and one state is the adjoint of the other. The generalized Hellmann-Feynman theorem is proven to hold... 

We consider a molecule containing a certain number of electrons moving in the field of fixed nuclei (Bom-Oppenheimer approximation). Let H be the electronic Hamiltonian depending on a set of parameters X (1 k <3N — 5 or 3N - 6) specifying nuclear positions. If ift and u represent the exact normalized wave function and energy for a nondegenerate electronic state, the generalized Hellmann-Feynman theorem states that ... 

So it is seen that the joint utilization of the antisymmetry principle and of the formulas derived from the generalized Hellmann-Feynman theorem allows us to find a quantum-mechanical justification of the Lewis ideas (electron pairing and sharing) and to analyze the physical nature of the chemical bond. 

Such an ensemble generalized ground-state energy functional, E = E[N, v] = E[p[N. i l- represents the thermodynamic potential of the N, v -representation, with the corresponding generalized Hellmann-Feynman expression for its differential (see equations (17), (22) and (27)) ... 

Equation (14.68) is the generalized Hellmann-Feynman theorem. [For a discussion of the origin of the Hellmann-Feynman and related theorems, see J. I. Musher, Am. J. Phys.,34,267 (1966).]... 

EXAMPLE Apply the generalized Hellmann-Feynman theorem to the one-dimensional harmonic oscillator with A taken as the force constant. 

Use the generalized Hellmann-Feynman theorem to find (pj) for the onedimensional harmonic-oscillator stationary states. Check that the result obtained agrees with the virial theorem. 

For a bound stationary state, the generalized Hellmann-Feynman theorem is dE /dX = f tjt dH/d ) dT, where A is a parameter in the Hamiltonian. (In case of degeneracy, i/r must be a correct zeroth-order wave function for the perturbation of changing A.) Taking A as a nuclear coordinate, we are led to the Hellmann-Feynman electrostatic theorem, which states that the force on a nucleus in a molecule is the sum of the electrostatic forces exerted by the other nuclei and the electron charge density. 

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 ... 

Equation (2.25) follows directly from Eq. (2.7), and represents a form of the generalized Hellmann-Feynman theorem (2.1) for the Coupled-cluster method. 

It is called the generalized Hellmann-Feynman theorem because Hellmann (1937) and Feynman (1939) considered originally the changes in energy due to a change in the geometry. 

This is a special case of the more general Hellmann-Feynman theorem, which we will encounter again in chapter 5. 

This is just the expectation or average value of the interaction operator, 6h, taken over the electron density function of the isolated molecule, as would have been anticipated from perturbation theory. This is a simple, powerful and extremely general result, which provides a basis for discussion of all atomic and molecular properties that involve a first-order response, and is applicable even when we do not possess exact wavefunc-tions it is usually referred to as a generalized Hellmann-Feynman... 

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