Hellmann-Feynman theorem force

This, the well-known Hellmann-Feynman theorem [128,129], can then be used for the calculation of the first derivatives. In nonnal situations, however, the use of an incomplete atom-centered (e.g., atomic orbital) basis set means that further terms, known as Pulay forces, must also be considered [130]. 

The Hellmann-Feynman theorem demonstrates the central role of p, the electron density distribution, in understanding forces in molecules and therefore chemical bonding. The main appeal and usefulness of this important theorem is that it shows that the effective force acting on a nucleus in a molecule can be calculated by simple electrostatics once p is known. The theorem can be stated as follows ... 

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... 

Perhaps of greater interest to us are results derived by the same authors71 that relate surface and bulk electronic properties of jellium. Considering two jellium slabs, one extending from —L to -D and the other from D to L, they calculated the force per unit area exerted by one on the other. According to the Hellmann-Feynman theorem, this is just the sum of the electric fields acting... 

In an ab initio simulation, the electronic structure problem is solved for each nuclear configuration, and forces are computed using the Hellmann-Feynman theorem. 

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. 

The electrostatic Hellmann-Feynman theorem states that for an exact electron wave function, and also of the Hartree-Fock wave function, the total quantum-mechanical force on an atomic nucleus is the same as that exerted classically by the electron density and the other nuclei in the system (Feynman 1939, Levine 1983). The theorem thus implies that the forces on the nuclei are fully determined once the charge distribution is known. As the forces on the nuclei must vanish for a nuclear configuration which is in equilibrium, a constraint may be introduced in the X-ray refinement procedure to ensure that the Hellmann-Feynman force balance is obeyed (Schwarzenbach and Lewis 1982). 

Equation (7) is the famous Hellmann-Feynman theorem which allows the full set of quantum-many-body forces to be calculated which can then be used to optimize the atomic geometry or to study the dynamics of the atoms by integrating the Newtonian equations of motion,... 

If the Hellmann-Feynman theorem is to be valid for forces on nuclei, the Coulomb cusp condition must be satisfied. However, if the nuclei are displaced, the orbital Hilbert space is modified. Hurley [179] noted this condition for finite basis sets, and introduced the idea of floating basis functions, with cusps that can shift away from the nuclei, in order to validate the theorem for such forces. 

Before turning to many-electron molecules, it is useful to ask Where does the energy of the chemical bond come from In VB theory it appears to be connected with exchange of electrons between different atoms but in MO theory it is associated with delocalization of the MOs. In fact, the Hellmann-Feynman theorem (see, for example, Ch.5 of Ref.[7]) shows that the forces which hold the nuclei together in a molecule (defined in terms of the derivatives of the total electronic energy with respect to nuclear displacement) can be calculated by classical electrostatics, provided the electron distribution is represented as an electron density P(r) (number of electrons per unit volume at point r) derived from the Schrodinger wavefunction k. This density is defined (using x to stand for both space and spin variables r, s, respectively) by... 

If electronic densities, in particular, if density matrices of large molecules are available, then an important variant of the Hellmann-Feynman theorem can be used for the computation of forces acting upon various nuclei, to be used for macromolecular geometry optimization. 

In many approximate methods, the error of calculated Hellmann-Feynman forces is significant. Following the introduction of the force method of direct, analytic differentiation of Hartree-Fock and related approximate energies by Pulay and by Pulay and Meyer," "the Hellmann-Feynman theorem is rarely used in computational applications. Note, however, that the Hellmann-Feynman theorem still plays a prominent role in studying various special problems." " ... 

Up till now, we have not specified a way to calculate the forces acting on the particles in our extended system. For the ions, the forces are computed using the Hellmann-Feynman theorem, which states that if E(r) is a stationary wave-function of the system, then... 

The main advantage of a plane wave basis set, in view of Molecular Dynamics, is the independence of the basis set elements with respect to the ionic positions. [Ill] As a result, the Hellmann-Feynman theorem can be applied straightforwardly, without additional so-called Pulay terms arising from a basis set that would be dependent on the nuclei positions. The forces on the ions will be calculated at virtually no extra-cost. There is also no Basis Set Superposition Error for the same reasons. Another advantage of plane wave basis sets is that their quality depends only on the number of wave-vectors considered ( cutoff , see later) it is thus easier both to compare results and to make convergence studies with only one number defining the quality of the basis set. Finally, on the computional side, plane wave basis sets have... 

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). 

In linear variational problems, one way of satisfying Hurley s conditions is to make the basis set closed with respect to the differential operators d/dp. Such a basis set is in principle infinite. Practically, however, the Hellmann-Feynman theorem will be approximately satisfied if, for each significantly populated basis function x, its derivatives with respect to the orbital centers, X, x are included in the basis set (Pulay, 1969). The use of augmented basis sets in conjunction with the Hellmann-Feynman theorem was considered by Pulay (1969, 1977) but dismissed as expensive. Recently, Nakatsuji et al. (1982) have recommended such a procedure. However, an analysis of their procedure (Pulay, 1983c Nakatsuji et al., 1983) reveals that it is not competitive with the traditional gradient technique. Much of the error in the Hellmann-Feynman forces is due to core orbitals. Therefore, methods based on the Hellmann-Feynman theorem presumably work better for effective core... 

From the chemical point of view, we must say these equations are not tractable and provide no useful information. In common, the study carried out by many authors (Salem, 1963b Byers-Brown, 1958 Byers-Brown and Steiner, 1962 Bader, 1960b Murrell, 1960 Berlin, 1951 Ben-ston and Kirtman, 1966 Davidson, 1962 Benston, 1966 Bader and Bandrauk, 1968b Kern and Karplus, 1964 Cade et al., 1966 Clinton, 1960 Phillipson, 1963 Empedocles, 1967 Schwendeman, 1966) on the force constants is based on the application of the virial and the Hellmann-Feynman or the electrostatic theorems. In particular, the Hellmann-Feynman theorem provides the expression for ki which relates the harmonic force constant to the properties of molecular charge distribution p(r), i.e., it follows (Salem, 1963b) that... 

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