The hypervirial theorem

As an extension of Noether s theorem to quantum mechanics, the hypervirial theorem [101] derives conservation laws from invariant transformations of the theory. Consider a unitary transformation of the Schrodinger equation, U(H — F)T = U(H — = 0, and assume the variational Hilbert space closed under a 

2 Hellmann-Feynman and virial theorems 4.2.1 Generalized Hellmann-Feynman theorem 

As an example, suppose for an A-electron system that energy E is approximated by an orbital functional [ / , ], which depends on one-electron orbital wave functions / , and on occupation numbers n, through a variational A-electron trial wave function A momentum displacement is generated by U = cxp(- 7r D), where D = r - In (he momentum representation of the orbital wave functions, 

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A second method of constrained variation involves the hypervirial theorems. The virial theorem is really a special case of a more general set of hypervirial theorems characterized by the equation... 

Chong and Rasiel52 later compared the method of Byers Brown with that of Rasiel and Whitman, and found that the Byers Brown formulation was the best for practical purposes of computation. Further work has been done by Chong53 and by Chen,64 who introduces weighting operators which are related to the hypervirial theorems but form stronger constraints. 

J2iW= he total force acting on the electrons. The hypervirial theorem implies that the mean value of this force vanishes for an isolated system, = 0. 

If the Hilbert space of the variational basis set is invariant under scale transformation, then the hypervirial theorem implies... 

Equations (75) and (76) yield the nuclear contribution to the average force and torque on the electron distribution in the presence of external radiation. If the hypervirial theorems for R, P, and L, (i.e., the momentum, force, and torque theorems) are satisfied, then... 

Another set of sum rules for the static electromagnetic shieldings is provided by the virial theorem [9a, 41, 42], that is, the hypervirial theorem... 

Further progress in obtaining a general principle from this expression for Q] is not possible for an arbitrary region of space. We now demonstrate that this result is transformed into the atomic statement of the hypervirial theorem when the subsystem 1 is restricted to one which satisfies a particular variational constraint. We shall introduce the variational constraint in terms of a trial function (f> and Show how the constraint, which limits the variations to a particular class of subsystems, can be applied from the beginning of the variation process, before the point = tjjis attained. The following conditions are to be fulfilled in the constrained variation ... 

Using the expression derived previously for a subsystem statement of the hypervirial theorem, eqn (5.43), we arrive at the atomic statement of the... 

Because of the presence of the surface term in eqn (6.2), the hypervirial theorem for a subsystem leads to important physical results which have no counterpart for the total system. It is our immediate goal to use eqn (6.3) to... 

The atomic statement of the principle of stationary action, eqn (6.3), yields a variational derivation of the hypervirial theorem for any observable G, a derivation which applies only to a region of space H bounded by a surface satisfying the condition of zero flux in the gradient vector field of the charge density,... 

The opening section of this chapter stressed the importance of the presence of the surface integral in the hypervirial theorem for an open system, eqn (6.2). Unlike the theorem for a total system, eqn (6.4), in which case the average of the commutator of any observable G with the Hamiltonian H vanishes, the corresponding result for an atom in a molecule is proportional to the flux in the effective single-particle vector current density of the property G through the atomic surface. As a result, the hypervirial theorem plays an important role in determining the properties of an atom in a molecule. It also enables one to relate an atomic property to a sum of bond contributions, as is now demonstrated. 

On the other hand, even for approximate wavefunctions which do not fulfill the hypervirial theorem, the CTOCD technique guarantees full elimination of terms to third order in the shift of origin, and partial cancellation of terms that depend on it linearly and quadratically. 

Show that the hypervirial theorem follows from Eq. (7.112). 

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