Hypervirial relation

This is connnonly used to measure the temperature in a MD simulation. Less well known is the hypervirial relation... 

Equations (87)-(112) for the molecular polarizabilities and susceptibilities have been obtained assuming the length and angular momentum gauges see Eqs. (35)-(37), (60), and (61). One could alternatively use the first-order Hamiltonians (35 ), (35"), and (37 ) in (57) and (59), defining molecular tensors in different formalisms. This amounts to introducing the off-diagonal hypervirial relations, which are consistent with Eqn. (64) ... 

Allowing for the hypervirial relations (115) and (116) and for the identities (117) and (122), one can partition the nuclear shieldings into atomic terms, in much the same way as for polarizabilities and susceptibilities. Thus... 

By means of these formulas and the hypervirial relations, we can prove that... 

In the usual consideration neglecting the possible degeneracy of the interacting molecules, the simplifications of the calculation of (t2) can be achieved by employing the hypervirial relation (Hirschfelder, 1960)... 

Aa.E. Hansen, T.D. Bouman, Hypervirial relations as constraints in calculations of electronic excitation properties The random phase approximation in configuration interaction language. Mol. Phys. 37 (1979) 1713. 

The HF theorem is satisfied for a given diatomic molecule at a given internuclear distance, as proved by Hirschfelder and Coulson (1962), if the set of so-called hypervirial relations corresponding to all the parameters AJ occurring in H, of the type ([H, W,]> = 0 with Hermitian operators WJ, holds. These hypervirial relations are an alternative expression of the floating or stable condition, which will be discussed later. As expected from this viewpoint, the virial and HF theorems have an intimate relation (Frost and Lykos, 1956). 

There exist a large number of different hypervirial relations (see one method of generation of such relations in [43]), but the best known is the virial theorem. We will consider this relation in the next subsection. 

The possibility of using the virial/hypervirial relations for approximate wavefunctions depends on the form and nature of the classes of approximate functions. The general theory is presented in [48]. For the use of the virial theorem for confined systems, see reference [49]. 

Frohlich s relations were also used in a slightly modified form of the perturbation theory power series in the dimensionless parameter (R — Ro)/Ro [57], In its current form, derivatives like 9rE(R) are usually derived on the basis of commutation relations as a consequence of some hypervirial relations in R3 (see [44-46]). It is clear that these derivatives appear as the boundary terms in the usual integral relations (see also Sections 3.2 and 3.3). 

Several important relations can be derived from the off-diagonal hypervirial relation (Hansen and Bouman, 1979), if one chooses an operator P that does not commute with the Hamiltonian The most important one is obtained for P = O, which is a cartesian component of the sum of the position operators of all electrons defined as ... 

Finally, using the off-diagonal hypervirial relation, Eq. (3.66) we arrive at the desired sum-over-states expression in Eqs. (5.99)-(5.101)... 

Hint Replace the gauge origin Rao in Eq. (5.44) and Eq. (5.85) by Rao + D, isolate the terms that depend on the arbitrary change D in the gauge origin and show that these terms cancel. Furthermore you might want to use the hypervirial relation Eq. (3.66) and the fact that the set of excited states is complete, i.e. 

Exercise 5.12 Show that the sum of the paramagnetic contribution to the nuclear magnetic shielding tensor Eq. (5.85) and the CTOCD-DZ diamagnetic contribution Eq. (5.115) is independent of the gauge origin Rao without making use of the hypervirial relation, Eq. (3.66),... 

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