Generalized Hellmann-Feynman theorem

Accordingly, the Hellmann-Feynman theorem holds for approximate wave functions as well, provided that the condition (5) or (6) holds for each wave function parameter A.. As an example of the contrary, the Hellmann-Feynman theorem generally does not hold for derivatives with respect to nuclear coordinates as basis functions usually follow the nuclei [7]. 

M. Menon and R. E. AUen, New technique for molecular-dynamics computer simulations Hellmann- Feynman theorem and subspace Hamiltonian approach , Phys. Rev. B33 7099 (1986) Simulations of atomic processes at semiconductor surfaces General method and chemisorption on GaAs(llO) , ibid B38 6196 (1988). 

Rather than giving the general expression for the Hellmann-Feynman theorem, we focus on the equation for a general diatomic molecule, because from it we can leam how p influences the stability of a bond. We take the intemuclear axis as the z axis. By symmetry, the x and y components of the forces on the two nuclei in a diatomic are zero. The force on a nucleus a therefore reduces to the z component only, Fz A, which is given by... 

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. 

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... 

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. 

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 electrostatic theorem, as a special case of the Hellmann-Feynman theorem, has some limitations. These limitations are briefly reviewed helow, following the discussion of the connections between the Hellmann-Feynman theorem and general properties of potential energy hypersurfaces. ... 

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) ... 

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. 

We seek a solution of H for arbitrary A . In Section 4.4 we discuss the Hellmann-Feynman theorem, which gives us a general solution for any eigenstate. For now, however, we describe the Peierls mechanism, which gives us the dimerized, broken-symmetry ground state. 

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. 

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". 

The above gradient formula of Eq. (14.18) has been obtained by assuming that the wave function in question satisfies the formal Hellmann-Feynman theorem of Eq. (14.4) with respect to the second quantized Hamiltonian in the given basis set. Gradient formulae for diverse variational wave functions can be obtained as special cases of the general result of Eq. (14.18). In what follows we shall demonstrate how the familiar SCF gradient formula (Pulay 1969, Fletcher 1970, Pople et al. 1979) can be obtained from Eq. (14.18). The one-determinantal nature of the SCF wave function allows us to introduce some simplifications. Namely, as it was shown in Sect. 7.2 ... 

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. 

From the variational condition (3.9), (t) r satisfies a generalization of the time-dependent Hellmann-Feynman theorem, and if we consider in the Hamiltonian (3.2) an external perturbation bV(co) with amplitude s and periodicity T = In/co, we obtain... 

---