ARTICLE pubs.acs.org/JPCA
Local Virial and Tensor Theorems Leon Cohen* City University of New York, New York, New York 10065, United States ABSTRACT: We show that for any wave function and potential the local virial theorem can always be satisfied by choosing a particular expression for the local kinetic energy. In addition, we show that for each choice of local kinetic energy there are an infinite number of quasi-probability distributions which will generate the same expression. We also consider the local tensor virial theorem.
1. INTRODUCTION One of the major contributions of Bader1 was the development of the concept of the virial theorem in a region and its use to study the spatial partitioning of molecules. Bader considered regions of space where the total flux vanishes. A fundamental issue in these considerations is the concept of local kinetic energy. It has been shown that there are an infinite number of expressions for local kinetic energy, each one associated with different quasi-probability distributions, and that for a wide class of expressions virial partitioning holds.4 We show that one can always choose a local kinetic energy expression so that the local virial theorem is satisfied. We also discuss the local tensor virial theorem.3 In related works, we mention that Mazziotti, Parr, and Simons11 derived the conditions for the virial theorem to hold in a region. In the Ghosh, Berkowitz, and Parr7 formulation as a local thermodynamic theory, the local kinetic energy plays a central role. Also, various definitions of local kinetic energy have been used in developing kinetic equations by Ziff, Kac, and Uhlenbeck13 and Putterman.12 Dahl and Springborg6 considered the question of whether the virial theorem can be satisfied at each point. They showed that one particular definition does yield a local virial theorem for the ground state of the hydrogen atom. There are a number of qualitatively different ways that the virial theorem was formulated in classical mechanics. The first is to consider time averages of closed orbits and relate them to spatial averages over the orbits. The other viewpoint, the viewpoint we take here, is to deal with a joint distribution of position and momentum. For an arbitrary distribution one can define global, mixed, and conditional moments. There is no particular reason that there should be a relation between them. However, if the distribution function is of a particular form, for example, a function of the Hamiltonian only, then we would expect a relation between the moments. The virial theorem expresses a relation between the second moment of momentum and the expected value of the potential function. From a physical point of view, such distribution functions are the ones that are in equilibrium; that is, the expected value of functions does not change in time. In addition, one could have a local equilibrium where local quantities are the conditional moments of the distribution. We also point out that for global quantities that are functions of position or momentum only the marginal r 2011 American Chemical Society
distributions are needed for their calculation average. The situation in quantum mechanics is quite different in certain aspects but also has considerable similarities. We describe some of these issues. Stationary states in quantum mechanics correspond to equilibrium in classical mechanics because for stationary states the probability of position and momentum are time independent. As for the global virial or tensor theorem in quantum mechanics, they are usually proven using operators but could be proven in the same manner as the classical case. In particular, if |ψ(r)|2 and |ϕ(p)|2 are the position and momentum quantum distribution functions, respectively, then considering them as marginals one can express the quantum virial theorem as Z 2
p2 jϕðpÞj2 dp ¼ 2m
Z r 3 ∇V jψðrÞj2 dr
ð1Þ
and similarly for the tensor virial theorem. However, the situation is quite different when we consider local quantities because in that case we need the conditional moments. However, to define conditional moments or local values we need a joint distribution of position and momentum. In quantum mechanics, we do not have such a joint distribution and hence we use what has come to be known as quasi-distributions. These are discussed in Section 3.
2. VIRIAL AND TENSOR THEOREMS The standard quantum mechanical virial theorem is 2 p 2 ∇2 ¼ Ær 3 ∇V æ 2m
ð2Þ
where expectation values are taken with respect to a stationary state. (We consider one particle in three dimensions. Generalization to the many-particle case is achieved by substituting the reduced first-order density matrix where appropriate. All integrals go from ∞ to ∞ unless otherwise stated.) If we were Special Issue: Richard F. W. Bader Festschrift Received: April 30, 2011 Revised: July 31, 2011 Published: August 24, 2011 12919
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dealing with a gas in equilibrium, one would also expect to have a local virial theorem. In the quantum mechanical case, by the local virial theorem, we shall mean 2KðrÞ ¼ r 3 ∇V
marginal distributions Z Fðr, pÞ dp ¼ jψðrÞj2
ð3Þ
Z
4
where K(r) is the local kinetic energy and V(r) is the local potential. The quantum mechanical tensor virial theorem is3 2ÆTij æ ¼ ÆVij æ
ð4Þ
where Tij and Vij are, respectively p2 ∂ ∂ 2m ∂xi ∂xj
ð5Þ
∂V ∂xj
ð6Þ
Tij ¼ Vij ¼ xi
The trace of eq 4 gives the standard virial theorem, eq 2. For the local tensor virial theorem, we write tij ðrÞ ¼ xi
∂V ∂xj
ð7Þ
where tij(r) is the local tensor and will be discussed in Section 5.
3. LOCAL MOMENTUM AND KINETIC ENERGY We summarize the concept of local momentum and local kinetic energy, the various expressions possible, and their relationship to joint quasi-probability distributions of position and momentum. There are an infinite number of possible definitions for K(r), the main requirement being that it yields to the proper quantum mechanical result when integrated over all space Z Z p2 ψ∇2 ψ dr ð8Þ KðrÞ dr ¼ 2m Using the quantum quasi-probability distributions, the local kinetic energy can be obtained from the distribution, as in the classical case4 Z 2 p Fðr, pÞ dp ð9Þ KðrÞ ¼ 2m where F(r,p) is the quasi-probability distribution of position and momentum. As there are an infinite number of distributions, there are an infinite number of expressions for K(r). In addition, as we will discuss, there are an infinite number of distributions which generate the same particular expression for local kinetic energy. All bilinear quasi-distributions can be generated from2,9 3 ZZZ 1 eiθ 3 r iτ 3 p þ iθ 3 u Φðθ, τÞψ Fðr, pÞ ¼ 4π2 1 1 ð10Þ u τp ψ u þ τp du dτ dθ 2 2 where Φ(θ,τ) is called the kernel and determines the distribution and its properties. By choosing different Φ’s, particular cases are obtained. The best way to think of these distributions is to consider them as representations of the density matrix.5 Their importance and appeal is that they have classical like behavior, and they allow quantum mechanics to be formulated in a classical type phase space.2,9 They satisfy the quantum mechanical
ð11Þ
Fðr, pÞ dr ¼ jϕðpÞj2
ð12Þ
where ϕ(p) is the momentum wave function. To ensure eqs 11 and 12, we must choose the kernel, Φ(θ,τ), such that Φð0, τÞ ¼ Φðθ, 0Þ ¼ 1
ð13Þ
These distributions, in general, take on negative values and hence can not be considered a proper probability distribution. For that reason, they are sometimes called quasi-distributions. Any one of them is equivalent to the density matrix, and any one member can be transformed into another. As is the case with the density matrix, the particular choice should be motivated by convenience and possible insight into the specific problem at hand. Quantum mechanical averages can be calculated by phase space integration since one can always set Z ZZ ψðrÞGðQ , PÞψðrÞ dr ¼ gðr, pÞFðr, pÞ dr dp ð14Þ where Q and P are the quantum mechanical operators of position and momentum and where G(Q,P) is a quantum operator that corresponds to the classical function g(r,p). For eq 14 to hold, we must take2,9 ZZ GðQ , PÞ ¼ γðθ, τÞΦðθ, τÞeiθ 3 Q þ iτ 3 P dθ dτ ð15Þ where γðθ, τÞ ¼
1 4π2
3 ZZ
gðr, pÞeiθ 3 r þ iτ 3 p dr dp
ð16Þ
3.1. Local Momentum. Consider the local momentum which we may define as the first conditional moment10 Z Æpær ¼ pFðr, pÞ dp ð17Þ
Using eq 10 one calculates that10 Æpær ¼
1 4π2
2 ZZ
eiθ 3 ðu rÞ ½jψðuÞj2 h1 ðθÞ
þ mjðuÞΦðθ, 0Þ du dθ
ð18Þ
where j(r) is the standard quantum mechanical current jðrÞ ¼
p ðψ∇ψ ψ∇ψÞ 2mi
ð19Þ
and h1 ðθÞ ¼ ∇τ Φðθ, τÞjτ ¼ 0
ð20Þ
Now if we take h1(θ) = 0 and also use eq 13 then Æpær ¼ mjðrÞ 12920
ð21Þ
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which shows that the first conditional momentum can be made to equal the quantum current. The total momentum is given Z Z Z p ψ∇ψ dr ð22Þ Æpæ ¼ Æpær dr ¼ m jðrÞ dr ¼ i
There are an infinite number of distributions that can generate any one of the above. The conditions on product kernel that generate the above expressions are 1 0 ΦA ð0Þ ¼ ip; 2
and agrees with the quantum mechanical result. 3.2. Local Kinetic Energy. To obtain the possible expressions for local kinetic energy, one substitutes eq 10 into eq 9 to obtain4
00
ΦB ð0Þ ¼
0
1 00 ΦC ð0Þ ¼ p2 4
0
ΦD ð0Þ ¼ 0
2
KðrÞ ¼
ΦC ð0Þ ¼ 0;
where j is the quantum mechanical current as defined above and h2 is given by h2 ðθÞ ¼ ∇2τ Φðθ, τÞjτ ¼ 0
ð24Þ
The first term in eq 9 is distribution independent, and the dependence of the second term on the distribution is only by way of the first and second partial derivatives of the kernel, Φ(θ,τ). We note that if we use the fact that ∇2 jψj ¼ ψðrÞ∇2 ψðrÞ þ ψðrÞ∇2 ψðrÞ þ 2j∇ψj
2
ð25Þ
the local kinetic energy may be written as KðrÞ
p2 1 j∇ψj2 ∇2 jψj 4 2m " # 3ZZ 2 1 iθ 3 ðu rÞ jψðuÞj h2 ðθÞ þ ih1 ðθÞ 3 jðuÞ du dθ e 2π 2m ¼
ΦP ðθ, τÞ ¼ Φðθ 3 τÞ
ð27Þ
Equation 23 reduces to KðrÞ ¼
p2 00 ½1 4ΦP ð0Þ=p2 ∇2 jψj2 8m 0
þ ðp2 =2mÞj∇ψj2 þ ΦP ð0Þ∇ 3 j
ð28Þ
where the primes denote differentiation with respect to the argument of ΦP. Some expressions for local kinetic energy which have been used in the literature are p2 2 ψ ∇ψ KA ¼ 2m
ð29Þ
p j∇ψj2 2m 2
KB ¼
KC ¼
p2 2 ½ψ ∇ ψ þ ψ∇2 ψ 4m
2
KD ¼
ð30Þ
p 1 j∇ψj2 ∇2 jψ2 j 4 2m
ð31Þ
ð35Þ ð36Þ
We emphasize that for any particular expression for local kinetic energy there are an infinite number of distributions which will yield it. That is so since there are an infinite number of different kernels which have identical h1 and h2. Consider for example the expression given by KD. For the Wigner distribution Φðθ, τÞ ¼ 1
ð37Þ
and hence it is clear from eq 36 that one can obtain KD from it. However, there are an infinite number of other distributions which yield KD. To obtain them, one chooses kernels so that its first and second derivative are equal to zero at zero. For example, the distributions obtained by using any one of the following kernels 1 2 2 Rx 2
ð38Þ
ΦðxÞ ¼ 1 þ sin Rx Rx
ð39Þ
ΦðxÞ ¼ e1=2R x
ð40Þ
ΦðxÞ ¼ cos Rx þ
ð26Þ For the subclass of kernels, called product kernels, where Φ(θ,τ) is a function of θ 3 τ
ð34Þ
00
ΦD ð0Þ ¼ 0;
ð23Þ
ð33Þ
1 2 p 4
0
ΦB ð0Þ ¼ 0;
2 p 2 2 ½ψ ðrÞ∇ ψðrÞ þ ψðrÞ∇ ψðrÞ 2j∇ψj 8m " # 3ZZ 2 1 iθ 3 ðu rÞ jψðuÞj e h2 ðθÞ þ ih1 ðθÞ 3 jðuÞ du dθ 2m 2π
1 00 ΦA ð0Þ ¼ p2 4
2 2
þ Rx
Rx
where R is an arbitrary constant and x = θτ will all yield the same expression for local kinetic energy as the Wigner distribution because they satisfy eq 36. Hence from a particular favorite choice for local kinetic energy, one distribution cannot be favored since there are an infinite number which give the same expression. A procedure to generate an infinite number of kernels which have the identical expression for local kinetic energy is as follows. Chose any Φ(θ,τ) which satisfies eq 13 and for given h1(θ) and h2(θ) form the new kernel8 Φnew ðθ, τÞ ¼ Φðθ, τÞ þ τ½h1 ðθÞ ∇τ Φðθ, τÞjτ ¼ 0 þ
1 2 τ ½h2 ðθÞ ∇2τ Φðθ, τÞjτ ¼ 0 2
ð41Þ
This assures that Φnew will satisfy eqs 20 and 24. In this way, we generate an infinite number of kernels and hence an infinite number of distributions which will give the same expression for the local kinetic energy. For the case of product kernels where the local kinetic energy is determined by Φ0 (0) and Φ00 (0), choose an arbitrary ΦP(θτ) for a given Φ0 (0) and Φ00 (0) and form 0
Φnew ðθτÞ ¼ ΦP ðθτÞ þ θτ½Φ0 ð0Þ ΦP ð0Þ
ð32Þ
þ 12921
1 2 2 00 00 θ τ ½Φ ð0Þ ΦP ð0Þ 2
ð42Þ
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We now ask under what conditions can one find a product kernel to give the same expression for local kinetic energy as given by a particular h1 and h2. We obtain the conditions by substituting ΦP(θτ) for Φ(θ,τ) into eqs 20 and 24 which results in 0
θΦP ð0Þ ¼ h1 ðθÞ
ð43Þ
00
θ2 ΦP ð0Þ ¼ h2 ðθÞ
5. LOCAL TENSOR VIRIAL THEOREM We define the local tensor tij(r) as the local value of pipj Z 1 tij ðrÞ ¼ ð50Þ pi pj Fðr, pÞ dp 2m Substituting eq 10 into 50, one obtains that tij ðrÞ ¼ " p2 ∂ ∂ ∂ ∂ ψ ðrÞ ψðrÞ þ ψðrÞ ψ ðrÞ 8m ∂xi ∂xj ∂xi ∂xj
ð44Þ
Hence we must have that h1 ðθÞ ∼ θ
ð45Þ
h2 ðθÞ ∼ θ2
ð46Þ
∂ ∂ ∂ ∂ ψ ðrÞ ψðrÞ ψ ðrÞ ψðrÞ ∂xi ∂xj ∂xj ∂xi " # 3 ZZ 1 1 iθ 3 ðu rÞ ∂ ∂ e Φðθ, τÞ jψðuÞj2 du dθ 2π 2m ∂τi ∂τj τ¼0 3 ZZ 1 i eiθ 3 ðu rÞ 2π 2 8 9 " #
