The Hellmann-Feynman Theorem

For example, in the hydrogen molecule-ion in its equilibrium configuration we may say that the electron is distributed in a way equivalent to having 3/7 of the electron spherically distributed about each nucleus and 1/7 at the midpoint between the two nuclei, giving a force of attraction by the electron for each nucleus that balances the force of repulsion by the other nucleus. 

FIGURE 14.3 Potential energy along the internuclear axis for electronic motion in H 2 at an intermediate internuclear distance. 

Consider a system with a time-independent Hamiltonian H that involves parameters. An obvious example is the molecular electronic Hamiltonian (13.5), which depends parametrically on the nuclear coordinates. However, the Hamiltonian of any system 

The force constant is a parameter, as is the mass m. Although ft is a constant, we can consider it as a parameter also. The stationary-state energies E are functions of the same parameters as For example, for the harmonic oscillator 

The stationary-state wave functions also depend on the parameters in H. We now investigate how E varies with each of the parameters. More specifically, let A be one of the parameters we ask for dEJdX, where the partial derivative is taken with all other parameters held constant. 

The integral in (14.57) is a definite integral over all space, and its value depends parametrically on A, since 7 and i/ depend on A. Provided the integrand is well behaved, we can find the integral s derivative with respect to a parameter by differentiating the integrand with respect to the parameter and then integrating. (Recall Problem 8.11b.) Thus 

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  

The force on a nucleus in a molecule is the sum of the Coulombic forces exerted by the other nuclei and by the electron density distribution p. 

The only forces operating in a molecule are electrostatic forces. There are no mysterious quantum mechanical forces acting in molecules. 

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 

Consider a system with a time-independent Hamiltonian H that involves parameters. An obvious example is the molecular electronic Hamiltonian (13.5), which depends parametrically on the nuclear coordinates. However, the Hamiltonian of any system contains parameters. For example, in the one-dimensional harmonic-oscillator Hamiltonian operator - f /2m) (f/dx ) + kx, the force constant is a parameter, as is the mass m. Although is a constant, we can consider it as a parameter also. The stationary-state energies E are functions of the same parameters as H. For example, for the harmonic oscillator 

The parameter A will occur in the kinetic-energy operator as part of the factor multiplying one or more of the derivatives with respect to the coordinates. For example, taking A as the mass of the particle, we have for a one-particle problem 

Equation (14.123) is the (generalized) Hellmann-Feynman theorem. [For a discussion of the origin of the Hellmann-Feynman and related theorems, see J. 1. Musher, Am. J. Phys., 34, 267 (1966).] 

Many of the theorems for exact wave functions hold also for approximate wave functions that are variationally determined. We consider here the Hellmann-Feyrman theorem, which states that the first-order change in the energy due to a perturbation may be calculated as the expectation value of the perturbation operator V [8]. This result is easily derived  

Here is the wave function associated with H + aV and 1 ) is the normalized, unperturbed wave function at a = 0. As an example, consider a molecule with the nuclei initially located at R/. Displacing the nuclei to R/ + /. we obtain the Hamiltonian 

Also the wave function is modified when the nuclei are displaced. However, the Hellmann-Feynman theorem shows that the contribution from the changes in the wave function vanishes for first derivatives  

The Hellmann-Feynman theorem thus greatly simplifies the evaluation of the first derivatives of variational energies. 

The Hellmann-Feynman theorem also holds for approximate wave functions provided they are optimized with respect to the ehanges induced by the perturbation  

It follows from the above MCSCF-based derivation that the Hellmann-Feynman theorem is fulfilled both for SCF and MCSCF wavefunctions since Eq. (5.15) yields, upon difierentiation with respect to a. 

It should, however, be pointed out that this result is a consequence of the fact that the SCF and MCSCF wavefunctions () have been optimized with respect to all variational parameters in 0 and that and in Eqs. (5.8) and (5.9) therefore vanish. If the orbital optimization is carried out using a limited number of the total set of variational parameters in 0 , the expansions in Eqs. (5.8) and (5.9) contain zeroth-order elements. The expansion of the total energy R ol) would then contain first-order terms in ot beyond 0 //, 0 and the Hellmann-Feynman theorem would therefore not be fulfilled. This is the case in a limited Cl calculation where the orbital variations are not considered explicitly [Eq. (5.9) contains zeroth-order terms]. Of course, the Hellmann-Feynman theorem is fulfilled in the full Cl limit, where the orbital optimization parameters are redundant. 

Before starting properly with perturbation theory we will first introduce in the next section the Hellmann Feynman theorem, which establishes a deep connection between the energy and molecular properties calculated as expectation values and that does not rely on perturbation theory. 

In Part II we will see that all molecular properties can be defined as derivatives of the energy with respect to the strength of external or internal perturbations. A theorem, which will become very useful in this context, is the Hellmann—Feynman theorem. 

Let us, for its derivation, consider a Hamiltonian H P) with eigenfunctions o(- )) and eigenvalues Eo P) that all depend on the general electromagnetic field P with components Pa-.- 

We assume further that the eigenfunctions are normalized for all values of Pa 

The Hamiltonian depends only explicitly on the field, which allows us to replace the total by the partial derivative of the Hamiltonian. In the second and third terms we can make use of the fact that o(. ) is an eigenfunction of H P), Eq. (3.3), and that the eigenvalues are real, which allows us to write 

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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)... 

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... 

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. 

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]... 

Recently, Romero and Andrews [1], and Lipinski [2] have shown that the calculated sum over states of a one-electron nonlinear optical property of a molecular system must vanish provided that the wave function employed satisfies the Hellmann-Feynman theorem. This statement applies, in particular, to the electric dipole polarizahility and, as a consequence, there must exist systems which exhibit, most prohahly in excited states, a negative polarizability. Several examples of atomic and molecular systems with negative polarizability can be found in Refs. [3-8]. In search for such systems we study the state of... 

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. 

By defining all these quantities as explicit functions of A, we can relate the density functional quantities to those more familiar from quantum chemistry. The exchange-correlation energy of density functional theory can be shown, via the Hellmann-Feynman theorem [38, 37], to be given by a coupling-constant average, i.e.. 

Using Eq. (10.1) to obtain, with the help of the Hellmann-Feynman theorem [74], the derivative of AE at constant electron density p, taking the nuclear charge Z. as variable, namely... 

Equation (10.2) is our starting point. Applying the Hellmann-Feynman theorem, we get... 

The Hellmann-Feynman theorem offers a convenient way to highlight the main features of chemical binding. By choosing the nuclear charges as parameters, it becomes possible to define the binding of each individual atom in a molecule without having recourse to an explicit partitioning of that molecule into atomic subspaces. 

Imagine that one could control the extent of electron-electron interactions in a many-electron system. That is, imagine a switch that would smoothly convert the non-interacting KS reference system to the real, interacting system. Using the Hellmann-Feynman theorem, one can... 

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... 

Beginning in the 1960s, Richard Bader initiated a systematic study of molecular electron density distributions and their relation to chemical bonding using the Hellmann-Feynman theorem.188 This work was made possible through a collaboration with the research group of Professors Mulliken and Roothaan at the University of Chicago, who made available their wave-functions for diatomic molecules, functions that approached the Hartree-Fock limit and were of unsurpassed accuracy. 

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