The Virial Theorem

M For further discussion of wave functions for the hydrogen molecule-ion see Introduction to Quantum Mechanics. 

In these equations TC is the average kinetic energy (always positive), V is the average potential energy, and W is the total energy, which is a constant. 

We now derive the virial theorem. Let H be the time-independent Hamiltonian of a system in the bound stationary state i/c 

Let A be a linear, time-independent operator. Consider the integral 

Equation (14.61) is the hypervirial theorem. [For some of its applications, see J. O. Hirschfelder, J. Chem. Phys., 33, 1462 (1960) J. H. Epstein and S. T. Epstein, Am. J. Phys., 30, 266 (1962).] In deriving (14.61), we used the Hermitian property of H. The proof that and are Hermitian, and hence that H is Hermitian, requires that ij/ vanish at 00 [see Eq. (7.17)]. Hence the hypervirial theorem does not apply to continuum stationary states, for which if/ does not vanish at °o. 

Equation (14.65) is the quantum-mechanical virial theorem. Note that its validity is restricted to bound stationary states. (The word vires is Latin for forces. In classical mechanics, the derivatives of the potential energy give the negatives of the force components. There is also a classical-mechanical virial theorem.) 

For certain systems the virial theorem takes on a simple form. To discuss these systems, we introduce the idea of a homogeneous function. A function /(xj, X2, , Xj) of several variables is homogeneous of degree n if it satisfies 

This chapter discusses two theorems that aid in understanding chemical bonding. We begin with the virial theorem. 

Equation (A10.9) shows that the values of the kinetic and potential energy for the Is electron are related by a factor of -2. This result is an example of the virial theorem. The virial theorem is a very general concept which says that for a particle moving in a potential that follows a power law, such as the Coulomb potential, the average kinetic and potential energy will be linked by a simple numerical factor. 

For a power law in which U = ar , with a the proportionality constant, the virial theorem states 

This very powerful idea is valid for both classical and quantum systems. In cases described by quantum mechanics it only works when we have correctly identified a stationary state of the system, and so can be a useful test of results. 

Equation (A10.9) shows that (T) = - U), which is the correct result for the Coulomb power law n = —1). That the virial theorem is obeyed is also confirmation that the exponent in the 5) function (Equation (A 10.3)) has the optimum form we are correct to use exp( -r) rather than some other radial decay, such as exp( r) with C 7 1, as will be checked in Problem AlO.l. The electron is distributed as a function of r, so the decay constant affects the averaging process and so is important in calculating the expectation values of the energies. 

Problem AlO.l We can use the virial theorem to obtain the exponential decay constant in the Is orbital function. To do this, consider the general form of a trial 5-function with a decay constant C and then we try to prove that only f = 1 gives a wavefunction that satisfies the virial theorem. The Greek letter zeta is widely used to describe the decay constants for basis sets in computational chemistry, and so we have adopted that convention here. The normalized trial function 5u) will be 

In this chapter, we discuss different types of classical mechanical atomic interaction and forms of the potential energy U in non-ideal gases, for which the configurational integral Z and the thermodynamic properties can be determined exactly. We begin with the virial theorem, which provides a bridge to numerical simulations for computing equations of state for systems whose interactions are not amenable to analytical calculations. 

Consider a system with N particles in volume V with constant temperature T. There are 6N degrees of freedom that define the microscopic states of this system  

We can define a new set of variables, that do not distinguish between coordinates and momenta, as follows  

This is the general mathematical representation of the virial theorem, applicable to all microscopic degrees of freedom. 

The virial theorem was first stated by Clausius in 1870 for the expectation value of the product between the position of each particle and the force acting on it. Indeed, substituting in Eq. 8.15 for the position of a particle, and using Hamilton s equation of motion, yields 

Using V)e vT o = 0, this operator generates a scale transformation 

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

The force acting on nucleus a is Fa = —from Fqs. (4.1) and (4.2), the general virial theorem for a molecule is 

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The pressure follows from the virial theorem, or from the characteristic thennodynamic equation and the PF. It is given by... 

The pressure is usually calculated in a computer simulation via the virial theorem ol Clausius. The virial is defined as the expectation value of the sum of the products of the coordinates of the particles and the forces acting on them. This is usually written iV = X] Pxi where x, is a coordinate (e.g. the x ox y coordinate of an atom) and p. is the first derivative of the momentum along that coordinate pi is the force, by Newton s second law). The virial theorem states that the virial is equal to —3Nk T. 

The energy obtained from a calculation using ECP basis sets is termed valence energy. Also, the virial theorem no longer applies to the calculation. Some molecular properties may no longer be computed accurately if they are dependent on the electron density near the nucleus. 

In spite of its simplicity and the visual similarity of this equation to Eq. (7), we would like to note that Eq. (11) leads to a nontrivial thermodynamics of a partially quenched system in terms of correlation functions, see, e.g.. Ref. 25 for detailed discussion. Evidently, the principal route for and to the virial theorem is to exploit the thermodynamics of the replicated system. However, special care must be taken then, because the V and s derivatives do not commute. Moreover, the presence of two different temperatures, Pq and P, requires attention in taking temperature derivatives, setting those temperatures equal, if appropriate, only at the end of the calculations. 

In a simulation [19] the pressure tensor is obtained from the virial theorem [78]... 

The energy of a Is-electron in a hydrogen-like system (one nucleus and one electron) is —Z /2, and classically this is equal to minus the kinetic energy, 1/2 mv, due to the virial theorem E — —T = 1/2 V). In atomic units the classical velocity of a Is-electron is thus Z m= 1). The speed of light in these units is 137.036, and it is clear that relativistic effects cannot be neglected for the core electrons in heavy nuclei. For nuclei with large Z, the Is-electrons are relativistic and thus heavier, which has the effect that the 1 s-orbital shrinks in size, by the same factor by which the mass increases (eq. (8.2)). 

Any changes in the potential energy because of the Coulomb correlation must therefore also influence the kinetic energy. The virial theorem will be further discussed below. 

If a trial function 9 leads to a kinetic energy 1 and a potential energy Vx which do not fulfill the virial theorem (Eq. 11.15), the total energy (7 +Ei) is usually far from the correct result. Fortunately, there exists a very simple scaling procedure by means of which one can construct a new trial function which not only satisfies the virial theorem but also leads to a considerably better total energy. The scaling idea goes back to a classical paper by Hylleraas (1929), but the connection with the virial theorem was first pointed out by Fock.5 It is remarkable how many times this idea has been rediscovered and published in the modern literature. 

Let us first assume that (px is the exact solution. This implies that the variation principle must be fulfilled ior r] = 1, and substitution of this value into Eq. 11.24 leads then to the virial theorem in the form of Eq. 11.15. 

In the derivation above, we have included the kinetic energy of the nuclei in the Hamiltonian and considered a stationary state. In Eq. II.3, this term has been neglected, and we have instead assumed that the nuclei have given fixed positions. It has been pointed out by Slater34 that, if the nuclei are not situated in the proper equilibrium positions, the virial theorem will appear in a slightly different form. (A variational derivation has been given by Hirschfelder and Kincaid.11)... 

By multiplying Eq. 11.32 by 77 and by using Eq. 11.30 and Eq. 11.31, it is then easily checked that the virial theorem (Eq. 11.33) is satisfied for the scaled function internuclear distance R — rj xp. The distance R is here a simple function of p, and, after establishing the relationship in the form of a graph or a table, we can also solve the reverse problem of finding the properly scaled func-... 

We note that the virial theorem is automatically fulfilled in the Hartree-Fock approximation. This result follows from the fact that the single Slater determinant (Eq. 11.38) built up from the Hartree-Fock functions pk x) satisfying Eq. 11.46 is the optimum wave function of this particular form, and, since this wave function cannot be further improved by scaling, the virial theorem must be fulfilled from the very beginning. If we consider a stationary state with the nuclei in their equilibrium positions, we have particularly Thf = — Fhf, and for the correlation terms follows consequently that... 

Wigner s formula is open to criticism also on another point, since he assumes the existence of a stationary electron state where the density is so low that the kinetic energy may be neglected. This is in contradiction to the virial theorem (Eq. 11.15), which tells us that the kinetic energy can never be neglected in comparison to the potential energy and that the latter quantity is compensated by the former to fifty per cent. A reexamination of the low density case would hence definitely be a problem of essential interest. 

The reason for this complication of the theory is evident the truncated set may contain certain variable parameters, and, if these are carefully adjusted to render the best possible description of a specific state, they may become rather unsuitable for the description of another state. According to Section II.C(3), a truncated set should, e.g., always contain a scale factor as a variable parameter and, if this quantity is fitted to the ground state, it may give a basic set which is rather "out of scale for even the first excited state. Since the virial theorem is not satisfied for this state, the corresponding total energy may be comparatively poorly reproduced. This implies that in treating excited states, it is desirable to have reliable criteria for the accuracy of both energies and wave functions. 

If the basic set xpk is chosen complete, the virial theorem will be automatically fulfilled and no scaling is necessary. In such a case, the wave function under consideration may certainly be expressed in the form of Eq. III. 18, but, if the basis is chosen without particular reference to the physical conditions of the problem, the series of determinants may be extremely slowly convergent with a corresponding difficulty in interpreting the results. It therefore seems tempting to ask whether there exists any basic set of spin orbitals. which leads to a most "rapid convergency in the expansion, Eq. III. 18, of the wave function for a specific state (Slater 1951). 

In the case the calculations are based on a truncated set Wlf 2,. . . containing adjustable parameters, the A splitting is of particular importance, since it permits the investigator to use different values of these parameters for different eigenvalues Xk— the relation III.95 will anyway be valid. The scale factor rj is such a parameter, and the results in Section II.C(3) and III.D(lb) show that, by means of the A splitting, it is now possible to get the virial theorem exactly fulfilled for at least one of the eigenfunctions associated with each Xk. 

The results show that it is possible to improve the Hartree-Fock energy —2.86167 at.u. considerably by means of a simple correlation factor, but also that it is essential to scale the total function W properly to fulfil the virial theorem. The parameters in the best function u of the form of Eq. III. 121 are further given below ... 

We now consider more interesting properties that can be extracted in our approach which cannot be extracted in a standard X-ray charge analysis. For a system at equilibrium, the virial theorem gives the total energy as... 

In previous work, we have mainly used the DPM model to investigate the effects of the coefficient of normal restitution and the drag force on the formation of bubbles in fluidized beds (Hoomans et al., 1996 Li and Kuipers, 2003, 2005 Bokkers et al., 2004 Van der Floef et al., 2004), and not so much to obtain information on the constitutive relations that are used in the TFMs. In this section, however, we want to present some recent results from the DPM model on the excess compressibility of the solids phase, which is a key quantity in the constitutive equations as derived from the KTGF (see Section IV.D.). The excess compressibility y can be obtained from the simulation by use of the virial theorem (Allen and Tildesley, 1990). 

The first term on the left of Eq. (5.12) vanishes in the absence of external pressure, which leads to the Virial Theorem... 

In the absence of nuclear energy sources, a star contracts on a thermal timescale and radiates energy at the expense of gravitational potential energy. Since, by the Virial Theorem, the total energy... 

Parameters of dynamically hot galaxies , i.e. various classes of ellipticals and the bulges of spirals, generally lie close to a Fundamental Plane in the 3-dimensional space of central velocity dispersion, effective surface brightness and effective radius or equivalent parameter combinations (Fig. 11.10). This is explained by a combination of three factors the Virial Theorem, some approximation to... 

In chemistry, several properties such as enthalpy of formation, dipole moments, etc., are analyzed for molecules on the basis of an additivity approximation. The same was applied to Compton profiles by Eisenberger and Marra [14], who measured the Compton profiles of hydrocarbons and extracted bond Compton profiles by a least squares fitting. This also enabled an approximate evaluation of the energy of these systems from the virial theorem. 

Although the absence of nuclei-centric structure makes direct chemical interpretation difficult, the EMD does have some other advantages. For instance, it is related to energy via the virial theorem stated previously and carries the valence information around p = 0. The entire nature of EMD topography is fixed by that at p = 0, as described by the hierarchy principle. 

The relationship between the exchange potential of DFT and the corresponding energy functional is established through the virial theorem. The two are related via the following relationship derived by Levy and Perdew [23]... 

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