Hellmann Feynman theorem

For exact wavefunctions, the Hellmann-Feynman [18,19] theorem states that the derivative of the energy with respect to a nuclear co-ordinate x equals the expectation value of the derivative of the Hamiltonian. 

When the wavefunction is expanded, using expansion parameters c, this theorem still holds if dE/dc=0, or when dc/dx=0. The first is the case for completely optimised wavefunctions and the second for wavefunctions where some, or all, of the coefficients are frozen. This can be seen when we write the derivative of E with respect to x as a sum of two terms  

The first term contains the direct dependence on x, the second the dependence on x through c. When dE/dc-0 (optimised), or dc/dx= 0 (frozen coefficients) the second part disappears  

A useful expression for evaluating expectation values is known as the Hell-mann-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 

After W.H.E. Schwarz et al., Bunsen-Maga-zin (1999) 10, 60. Portrait reproduced from a painting by Tatiana Livschitz, courtesy of Professor Eugen Schwarz. 

Richard Philips Feynman (1919-1988), American physicist, for many years professor at the California Institute of Technology. His father was his first informal teacher of physics, who taught him the extremely important skill of independent thinking. Feynman studied at Massachusetts Institute of Technology, then in Princeton University, where he defended his Ph.D. thesis under the supervision of John Archibald Wheeler. 

In 1945-1950 Feynman served as a professor at Cornell University. A paper plate thrown in the air by a student in the Cornell cafe was the first impulse for Feynman to think about creating a new version of quantum electrodynamics. For this achievement Feynman received the Nobel prize in 1965, of. p. 14. 

From John Slater s autobiography Solid State and Molecular Theory , London, Wiley (1975)  

The Hellmann-Feynman theorem pertains to the rate of the change of E P)  

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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]. 

Bala, R, Lesyng, B., McCammon, J.A. Extended Hellmann-Feynman theorem for non-stationary states and its application in quantum-classical molecular dynamics simulations. Chem. Phys. Lett. 219 (1994) 259-266. 

Another way of obtaining molecular properties is to use the Hellmann-Feynman theorem. This theorem states that the derivative of energy with respect to some property P is given by... 

This relationship is often used for computing electrostatic properties. Not all approximation methods obey the Hellmann-Feynman theorem. Only variational methods obey the Hellmann-Feynman theorem. Some of the variational methods that will be discussed in this book are denoted HF, MCSCF, Cl, and CC. 

These expressions are only correct for wave functions that obey the Hellmann-Feynman theorem. Flowever, these expressions have been used for other methods, where they serve as a reasonable approximation. Methods that rigorously obey the Flellmann-Feynman theorem are SCF, MCSCF, and Full CF The change in energy from nonlinear effects is due to a change in the electron density, which creates an induced dipole moment and, to a lesser extent, induced higher-order multipoles. 

Variational wave functions thus obey the Hellmann-Feynman theorem. 

In such cases the expression from fii st-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, MP or CC), the Hellmann-Feynman theorem does not hold, and a first-order property calculated as an expectation value will not be identical to that obtained as an energy derivative. Since the Hellmann-Feynman theorem holds for an exact wave function, the difference between the two values becomes smaller as the quality of an approximate wave function increases however, for practical applications the difference is not negligible. It has been argued that the derivative technique resembles the physical experiment more, and consequently formula (10.21) should be preferred over (10.18). 

If the perturbation is a homogeneous electric field F, the perturbation operator P i (eq. (10.17)) is the position vector r and P2 is zero. As.suming that the basis functions are independent of the electric field (as is normally the case), the first-order HF property, the dipole moment, from the derivative formula (10.21) is given as (since an HF wave function obeys the Hellmann-Feynman theorem)... 

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). 

To obtain the Hellmann-Feynman theorem, we differentiate equation (3.67) with respect to X... 

Although the expectation value r )ni cannot be obtained from equation (6.70), it can be evaluated by regarding the azimuthal quantum number I as the parameter in the Hellmann-Feynman theorem (equation (3.71)). Thus, we have... 

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... 

Here, n(z z) is the charge density induced at z by a charged plane, of unit surface charge density, located at z, and pb is the bulk jellium density. The Hellmann-Feynman theorem was used. The full moment of n(z z) obeys... 

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. 

Heller equations, direct molecular dynamics, Gaussian wavepackets and multiple spawning, 399-402 Hellmann-Feynman theorem ... 

The adiabatic coupling matrix elements, Fy, can be evaluated using an off-diagonal form of the Hellmann-Feynman theorem... 

Recalling y(s), the adiabatic-to-diabatic transformation angle [see Eqs. (74) and (75)] it is expected that the two angles are related. The connection is formed by the Hellmann-Feynman theorem, which yields the relation between the 5 component of the non-adiabatic coupling term, x, namely, rs, and the characteristic diabatic magnitudes [13]... 

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