The Hellman-Feynman Theorem

The Hellman-Feynman theorem deserves a place in history and in this text but all attempts to make profitable use of the theorem failed because of the small print contained in the paragraph above. 

Note Pulay s mention of the Hellman-Feynman theorem. We have moved on since 1969, especially in the development and application of analytical methods for evaluating the gradients. 

Quantum mechanical models at different levels of approximation have been successfully applied to compute molecular hyperpolarizabilities. Some authors have attempted a complete determination of the U.V. molecular spectrum to fill in the expression of p (15, 16). Another approach is the finite-field perturbative technique (17) demanding the sole computation of the ground state level of a perturbated molecule, the hyperpolarizabilities being derivatives at a suitable order of the perturbed ground state molecule by application of the Hellman-Feynman theorem. 

How can one join an electronic structure calculation with a classical MD scheme In principle, this is possible in a straightforward manner - we can optimize the electronic wavefunction for a given initial atomic configuration (at time t=0) and calculate the forces acting on the atoms via the Hellman-Feynman theorem ... 

In a second approach of the reactivity, one fragment A is represented by its electronic density and the other, B, by some reactivity probe of A. In the usual approach, which permits to define chemical hardness, softness, Fukui functions, etc., the probe is simply a change in the total number of electrons of A. [5,6,8] More realistic probes are an electrostatic potential cf>, a pseudopotential (as in Equation 24.102), or an electric field E. For instance, let us consider a homogeneous electric field E applied to a fragment A. How does this field modify the intermolecular forces in A Again, the Hellman-Feynman theorem [22,23] tells us that for an instantaneous nuclear configuration, the force on each atom changes by... 

We consider a variation of the external potential 5vext(r) at constant electron number N. The formal expression of the energy variation due to this perturbation can be found by a direct application of the Hellman-Feynman theorem [22,23,26]. 

S referred to as the Hellman-Feynman theorem. It was widely used to investi-ate isoelectronic processes such as isomerizations X —> Y, barriers to internal Otation, and bond extensions where the only changes in the energy are due to hanges in the positions of the nuclei and so the energy change can be calculated tom one-electron integrals. 

Notice that the Hellman-Feynman theorem only applies to exact wavefunc- Mis, not to variational approximations. All the enthusiasm of the 1960s and jJTOs evaporated when it was realized that approximate wavefunctions them-jpves also depend on nuclear coordinates, since the basis functions are usually... 

This approach is based on some wide-ranging preconditions. In order to bridge the gap between microscopic molecular nature of a particle surface and macroscopic properties, a multi-scale approach covering several orders of magnitude of space and time is needed. On the most basic level quantum mechanics prevail. However, it is often possible by using the Hellman-Feynman theorem [3] to transfer the intrinsic quantum mechanical nature of surfaces to the physics... 

This expression can be easily and accurately calculated by using a DFT electronic structure code. First, one can make use of the Hellman-Feynman theorem to extract the gradient from Eq. (A.4). Second, one can use finite differences to estimate the gradient on the electronic Hamiltonian matrix elements following the explanation of the theory section above. 

In the TF limit when Z and N become very large, we know that n - 0 for the neutral atom case N=Z and hence /q(1)=0. March and Parr12 argue that this property must also imply /J(l)=/ (1)=/ (1)=0 and possibly also /i(l)=0, leaving p(Z, Z) Z-1/3, but again this has not been presently proved. One can also obtain Ken from the Hellman-Feynman theorem... 

Calculation of a dipole might be accomplished by taking the expectation value of the dipole moment operator. That result will be equivalent to the result obtained from invoking the strict definition of the dipole moment as a derivative in the case where the wavefunction obeys the Hellman-Feynman theorem [1,44] or, in general, where the wavefunction is completely variational. 

One very prominent development in DFT has been the coupling of electronic structure calculations (which, when the ground state is concerned, apply to zero temperature) with finite-temperature molecular dynamics simulations. The founding paper in this field was published by Carr and Parrinello in 1985 [13]. Carr and Parrinello formulate effective equations of motion for the electrons to be solved simultaneously with the classical equations of motion for the ions. The forces on the ions are calculated from first principles by use of the Hellman-Feynman theorem. An alternative to the Carr-Parrinello method is to solve the electronic structure self-consistently at every ionic time step. Both methods are referred to as ab initio molecular dynamics (AIMD) [14]. 

However, for the Hellman-Feynman theorem, (11), to be valid, the electronic density should be the one minimizing the total energy density functional... 

It is of special interest to find the configurations for which the electronic energy has its minimum. According to the Hellman-Feynman theorem quantum mechanical rule) this can be obtained from... 

Using the Hellman-Feynman theorem, the ground state electronic energy for the system is then... 

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