Hellmann-Feynman theorem Hamiltonian

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). 

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. 

Derivatives of the dipole moment with respect to Qj can be expressed within a Cartesian reference frame via a similarity transformation, introducing atomic polar tensors (APTs) [13, 14], The connection between the latter and the electric shielding is obtained by means of the Hellmann-Feynman theorem. Within the Born-Oppenheimer approximation and allowing for the dipole length formalism, the perturbed Hamiltonian in the presence of a static external electric field E is given by Eqs. (6) and (35). 

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,... 

As mentioned in section 1, the combination of the CI method and semiempirical Hamiltonians is an attractive method for calculations of excited states of large organic systems. However, some of the variants of the CI ansatz are not in practical use for large molecules even at the semiempirical level. In particular, this holds for full configuration interaction method (FCI). The truncated CI expansions suffer from several problems like the lack of size-consistency, and violation of Hellmann-Feynman theorem. Additionally, the calculations of NLO properties bring the problem of minimal level of excitation in CI expansion neccessary for the coirect description of electrical response calculated within the SOS formalism. 

This theorem is also valid for many variational wavefunctions, e.g. for the Hartree-Fock one, if complete basis sets are used. As only the one-electron part of the Hamiltonian depends on the nuclear coordinates, H is a one-electron operator, and the evaluation of the Hellmann-Feynman forces is simple. Because of this simplicity, there have been a number of early suggestions to use the Hellmann-Feynman forces for the study of potential surfaces. These attempts met with little success, and the discussion below will show the reason for this. It is perhaps fair to say that the main value of the Hellmann-Feynman theorem for geometrical derivatives is in the insight it provides, and that numerical applications do not appear promising. For other types of perturbations, e.g. for weak external fields, the theorem is widely used, however. For a survey, see a recent book (Deb, 1981). 

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 ... 

The Hellmann-Feynman theorem holds for the density-functional expression for Eu, and it is instructive to work through this. Writing the one-electron eigenvalues in terms of the Hamiltonian h in (9), (11) becomes ... 

The Parr theorem, regarded as the finite difference formula associated with the Hellmann-Feynman theorem, has some interesting consequences. In most quantum-chemical models, the difference of two Hamiltonians on the right-hand side is a one-electron operator. Consequently, at least in principle, in the calculation of the conformational energy difference the evaluation of complicated two-electron... 

A possible alternative approach to the calculation of forces is via the use of the Hellmann-Feynman theorem. If is an exact wavefunction of a Hamiltonian H with energy E then this theorem states that the derivative of E with respect to some parameter P can be written ... 

The quantum mechanical many-body nature of the interatomic forces is taken into account naturally through the Hellmann-Feynman theorem. Since the scheme usually uses a minimal basis set for the electronic structure calculation and the Hamiltonian matrix elements are parametrized, large numbers of atoms can be tackled within the present computer capabilities. One of the distinctive features of this scheme in comparison with other empirical schemes is that all the parameters in the model can be obtained theoretically. It is therefore very useful for studying novel materials where experimental data are not readily available. The scheme has been demonstrated to be a powerful method for studying various structural, dynamical, and electronic properties of covalent systems. 

It immediately follows from the hermitian character of the respective subsystem hamiltonians and their eigenvalue equations that these energy functions satisfy the following differential Hellmann-Feynman theorems ... 

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 Hellmann-Feynman theorem says that = ( ) (where H is the Hamiltonian that depends on parameter P). This is true, when... 

The electronic Hamiltonian is time-independent (no external electric field), but it changes with time if the nuclei are allowed to move. Hence, we have the following relation (if it obeys the Hellmann-Feynman theorem) ... 

The Hellmann-Feynman theorem says that (H means the Hamiltonian depending on... 

Note that Eq. (60) is not the Hellmann-Feynman theorem,to which it bears a formal resemblance, since in Eq. (60) it is not the Hamiltonian operator i e(x X) but rather the Hamiltonian matrix H (X) which is being differentiated. Indeed, from Eq. (49a)... 

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