The Virial and Hellmann-Feynman Theorems

Course: Molecular Quantum MechanicsYear: 2018/2019

VHFT

This chapter discusses two theorems that aid in understandingchemical bonding: the virial theorem and the Hellmann-Feynmantheorem, which are here presented after a preliminary introductionillustrating another theorem of quantum mechanics due toEherenfest.

VHFT

The Eherenfest Theorem

The Eherenfest theorem deals with the expectation values of theposition and momentum operators of a particle of mass m and theexpectation value of the force acting on it. In particular it relatesthe time derivative of 〈r〉 to 〈p〉 and the time derivative of 〈p〉 to〈F〉, where F = −∇V , i.e., the gradient of the scalar potential inwhich the particle moves. The theorem states

d

dt〈rx〉 =

1

m〈px〉

d

dt〈px〉 =

⟨−∂V∂x

⟩

VHFT

Eherenfest’s theorem can be proved to be a special case of a moregeneral relation of quantum mechanics, which connects the timederivative of the expectation value of any quantum mechanicaloperator A to the commutator of that operator and theHamiltonian of the system

d

dt〈A〉 =

1

i~

⟨[A, H]

⟩+

⟨∂A

∂t

⟩If the operator A does not contain time explicitly the last term inthe above equation will be vanishing.

VHFT

Let us derive the above equation within the Schrodinger picture,i.e., i~(∂Φ/∂t) = HΦ

d

dt〈A〉 =

d

dt

⟨Φ∣∣∣A∣∣∣Φ

⟩=

⟨(∂Φ

∂t

) ∣∣∣A∣∣∣Φ

⟩+

⟨Φ

∣∣∣∣∣(∂A

∂t

)∣∣∣∣∣Φ

⟩+

⟨Φ∣∣∣A∣∣∣ (∂Φ

∂t

)⟩

=

⟨(HΦ

i~

)∣∣∣A∣∣∣Φ

⟩+

⟨∂A

∂t

⟩+

⟨Φ∣∣∣A∣∣∣( HΦ

i~

)⟩

=1

i~

(⟨Φ∣∣∣A∣∣∣ HΦ

⟩−⟨HΦ

∣∣∣A∣∣∣Φ⟩)

+

⟨∂A

∂t

⟩=

1

i~

⟨Φ∣∣∣AH − HA

∣∣∣Φ⟩

+

⟨∂A

∂t

⟩=

1

i~

⟨[A, H]

⟩+

⟨∂A

∂t

⟩C.V.D.

VHFT

Setting A = rx one has

d

dt〈rx〉 =

1

i~

⟨[rx , H]

⟩Using [rx , H] =

i~mpx , see Levine Quantum Chemistry (5.9)

d

dt〈rx〉 =

1

m〈px〉

VHFT

Setting A = px one has

d

dt〈px〉 =

1

i~

⟨[px , H]

⟩Considering that

[px , H] = [px , T ] + [px , V ] = [px , V ] = −i~[∇x ,V ]

= −i~ (∇xV ·+V · ∇x − V · ∇x) = −i~∂V∂x

Therefored

dt〈px〉 =

⟨−∂V∂x

⟩and Ehrenfest’s theorem is proved.

VHFT

The Hypervirial Theorem

Let ψn and ψm be two bound stationary states of a system

Hψn = Enψn , Hψm = Emψm

Let A be a linear, time-independent operator. Consider the integral∫ψ∗n

[H, A

]ψm dτ =

⟨n∣∣∣HA− AH

∣∣∣m⟩ =⟨n∣∣∣HA

∣∣∣m⟩−⟨n ∣∣∣AH∣∣∣m⟩Since H is hermitian, the following hypervirial identities areobtained ∫

ψ∗n

[H, A

]ψm dτ = (En − Em)

⟨n∣∣∣A∣∣∣m⟩

∫ψ∗n

[H, A

]ψn dτ = 0

VHFT

Considering that

[H, rx ] = − i~mpx

[H, px ] = i~∂V

∂x

one obtains

〈n |rx |m〉 = − i~m

〈n |px |m〉En − Em

〈n |px |m〉 = i~⟨n∣∣∂V∂x

∣∣m⟩En − Em

which can be used to switch between the so called length, velocityand force formalisms.

VHFT

We now derive the virial theorem for a system containing nparticles from the hypervirial relation∫

ψ∗n

[H, A

]ψn dτ = 0

We choose A to be

n∑i

rαi pαi = −i~n∑i

rαi∂

∂rαi(α = x , y , z)

To evaluate[H, A

]we use[

H,n∑i

rαi pαi

]=

n∑i

{[H, rαi pαi

]± rαi Hpαi

}=

n∑i

rαi

[H, pαi

]+

n∑i

[H, rαi

]pαi

VHFT

[H,

n∑i

rαi pαi

]= i~

n∑i

rαi∂V

∂rαi− i~

n∑i

1

mip2αi

= i~n∑i

rαi∂V

∂rαi− 2i~T

Substitution of the last identity into the integral gives∫ψ∗n

{i~

n∑i

rαi∂V

∂rαi− 2i~T

}ψn dτ = 0

⟨ψn

∣∣∣∣∣n∑i

rαi∂V

∂rαi

∣∣∣∣∣ψn

⟩= 2

⟨ψn

∣∣∣T ∣∣∣ψn

⟩Using quantum-mechanical averages, we write⟨

n∑i

rαi∂V

∂rαi

⟩= 2 〈T 〉

which is the quantum-mechanical virial theoremVHFT

If V is a homogeneous function of degree g when expressed inCartesian coordinates, Euler’s theorem on homogeneous functiongives

n∑i

rαi∂V

∂rαi= gV

and the virial theorem simplify to

g 〈V 〉 = 2 〈T 〉

For a bound stationary state E = 〈T 〉+ 〈V 〉 we can write virialtheorem in two other forms

〈V 〉 =2E

g + 2

〈T 〉 =gE

g + 2

VHFT

For the H atom, V = −e2/(x2 + y2 + z2)1/2 is a homogeneousfunction of degree g = −1.

For a many-electron atom with spin-orbit interaction neglected, itcan be easily shown that V is still a homogeneous function ofdegree g = −1. Hence, for any atom

2 〈T 〉 = −〈V 〉 , (spin-orbit interaction neglected)

For a molecule in the Born-Oppenheimer approximation, V ≡ Vel

is a homogeneous function of degree g = −1 only when Vel is afunction of both the electronic and the nuclear Cartesiancoordinates.

In such a case, the molecular electronic virial theorem contains anextra term

2 〈Tel〉 = −〈Vel〉 −N∑I

RαI∂Eel

∂RαI

VHFT

Using

V = Vel + VNN , U(RαI ) = Eel(RαI ) + VNN

the virial theorem takes the form

2 〈Tel〉 = −〈V 〉 −N∑I

RαI∂U

∂RαI

VHFT

The Virial Theorem and Chemical Bonding

We now use the virial theorem to examine the changes inelectronic kinetic and potential energy that occur when a covalentchemical bond is formed in a diatomic molecule. For formation ofa stable bond, the U(R) curve must have a substantial minimum.At this minimum we have

dU

dR

∣∣∣∣Re

= 0

2 〈Tel〉|Re= − 〈V 〉|Re

〈Tel〉|Re= −U(Re)

〈V 〉|Re= 2U(Re)

At R =∞ we have the separated atoms

2 〈Tel〉|∞ = − 〈V 〉|∞〈Tel〉|∞ = −U(∞)

〈V 〉|∞ = 2U(∞)

VHFT

U(∞) is the sum of the energies of the two separated atoms. Forbonding, we have U(Re) < U(∞). Therefore,

〈Tel〉|Re− 〈Tel〉|∞ = U(∞)− U(Re) > 0

〈V 〉|Re− 〈V 〉|∞ = 2 [U(Re)− U(∞)] < 0

The decrease in potential energy is twice the increase in kineticenergy, and results from allowing the electrons to feel theattractions of both nuclei and perhaps from an increase in orbitalexponents in the molecule. The equilibrium dissociation energy is

De = U(∞)− U(Re) =1

2

[〈V 〉|∞ − 〈V 〉|Re

]

VHFT

Consider the behaviour of the average potential and kineticenergies for large R.

The interactions between atoms at large distances are called vander Waals forces. For two neutral atoms, at least one of which isin an S state, quantum-mechanical perturbation theory shows thatthe van der Waals force of attraction is proportional to 1

R7 , and thepotential energy behaves like

U(R) ≈ U(∞)− A

R6, (R large)

where A is a positive constant. This expression was first derived byLondon, and van der Waals forces between neutral atoms arecalled London forces or dispersion forces.

VHFT

VHFT

THE HELLMANN-FEYNMAN THEOREM

Consider a system with a time-independent Hamiltonian H thatinvolves parameters. An obvious example is the molecularelectronic Hamiltonian that depends parametrically on the nuclearcoordinates.

We begin with the Schrodinger equation Hψn = Enψn where theψn’s are the normalized stationary-state eigenfunctions. Because ofnormalization, we have

En =⟨ψn

∣∣∣H∣∣∣ψn

⟩∂En

∂λ=

∂

∂λ

⟨ψn

∣∣∣H∣∣∣ψn

⟩

VHFT

Provided the integrand is well behaved

∂En

∂λ=

∫∂

∂λ(ψ∗nHψn)dτ =

∫∂ψ∗n∂λ

Hψndτ +

∫ψ∗n

∂

∂λ(Hψn)dτ

The potential-energy operator is just multiplication by V , so

∂

∂λ(Vψn) =

∂V

∂λψn + V

∂ψn

∂λ

The parameter λ will occur in the kinetic-energy operator as partof the factor multiplying one or more of the derivatives withrespect to the coordinates. Since we can change the order of thepartial differentiations without affecting the result, we can write

∂

∂λ(Tψn) =

∂T

∂λψn + T

∂ψn

∂λ

where ∂T/∂λ is found by differentiating T with respect to λ justas if it were a function instead of an operator.

VHFT

Combining the two pieces, we can write

∂

∂λ(Hψn) =

∂H

∂λψn + H

∂ψn

∂λ

Therefore

∂En

∂λ=

∫∂ψ∗n∂λ

Hψndτ +

∫ψ∗n∂H

∂λψndτ +

∫ψ∗nH

∂ψn

∂λdτ

For the first integral we have∫∂ψ∗n∂λ

Hψndτ = En

∫∂ψ∗n∂λ

ψndτ

For the last integral we have∫ψ∗nH

∂ψn

∂λdτ =

∫∂ψn

∂λ(Hψn)∗dτ = En

∫ψ∗n∂ψn

∂λ

VHFT

Substitution gives

∂En

∂λ=

∫ψ∗n∂H

∂λψndτ + En

(∫∂ψ∗n∂λ

ψndτ +

∫ψ∗n∂ψn

∂λ

)∂En

∂λ=

∫ψ∗n∂H

∂λψndτ + En

(∂

∂λ

∫ψ∗nψndτ

)The wave function is normalized; hence

∂En

∂λ=

∫ψ∗n∂H

∂λψndτ

The last equation is the generalized Hellmann-Feynmantheorem.

The first-order perturbation theory result is a special case of theHellmann-Feynman theorem. The Hellmann-Feynman theoremwith En being the Hartree-Fock energy is obeyed by Hartree-Fock(as well as exact) wave functions.

VHFT

THE ELECTROSTATIC THEOREM

Hellmann and Feynman independently applied the theorem tomolecules taking λ as a nuclear Cartesian coordinate.

In the Born-Oppenheimer approximation, the electronicSchrodinger equation for a fixed nuclear configuration is

Hψel = (Tel + V )ψel = Uψel

whereV = Vel + VNN

VHFT

The Hamiltonian H depends on the nuclear coordinates asparameters. If xδ is the x coordinate of nucleus δ, the generalizedHellmann-Feynman theorem gives

∂U

∂xδ=

∫ψ∗el

∂H

∂xδψeldτel

The kinetic-energy part of H is independent of the nuclearCartesian coordinates. Hence

∂U

∂xδ=

∫ψ∗el

∂V

∂xδψeldτel

We have∂V

∂xδ=∂Vel

∂xδ+∂VNN

∂xδ

VHFT

In atomic units

Vel = −N∑α

n∑i

Zαriα

Vel = −N∑α

n∑i

Zα

[(xi − xα)2 + (yi − yα)2 + (zi − zα)2]1/2

∂Vel

∂xδ= −

n∑i

Zδ(xi − xδ)

[(xi − xδ)2 + (yi − yδ)2 + (zi − zδ)2]3/2

∂Vel

∂xδ= −

n∑i

Zδ(xi − xδ)

r3iδ

VHFT

VNN =N∑α

N∑β>α

ZαZβrαβ

VNN =N∑α

N∑β>α

ZαZβ

[(xα − xβ)2 + (yα − yβ)2 + (zα − zβ)2]1/2

∂VNN

∂xδ=

∂

∂xδ

N∑α 6=δ

ZαZδ

[(xα − xδ)2 + (yα − yδ)2 + (zα − zδ)2]1/2

∂VNN

∂xδ=

N∑α 6=δ

ZαZδ(xα − xδ)

[(xα − xδ)2 + (yα − yδ)2 + (zα − zδ)2]3/2

∂VNN

∂xδ=

N∑α 6=δ

ZαZδ(xα − xδ)

r3αδ

VHFT

Since ∂VNN/∂xδ does not involve the electronic coordinates andψel is normalized, after substitution in the integral we have

∂U

∂xδ= −Zδ

∫|ψel|2

n∑i

xi − xδr3iδ

dτel + Zδ

N∑α 6=δ

Zαxα − xδr3αδ

which can be written using the electron probability density functionas

∂U

∂xδ= −Zδ

∫ρ(r)

x − xδ|r− rδ|3

dr + Zδ

N∑α 6=δ

Zαxα − xδ|rα − rδ|3

The quantity −∂U/∂xδ can thus be viewed as the x component ofthe effective force on nucleus δ due to the other nuclei and theelectrons.

Actually, we have two corresponding equations for ∂U/∂yδ and∂U/∂zδ.

VHFT

If Fδ is the effective force on nucleus δ, then

Fδ = −∇δU = −i ∂U∂xδ− j

∂U

∂yδ− k

∂U

∂zδ

Fδ = −Zδ∫ρ(r)

rδ − r

|r− rδ|3dr + Zδ

N∑α6=δ

Zαrδ − rα|rα − rδ|3

Thus the effective force acting on a nucleus in a molecule can becalculated by simple electrostatics as the sum of the Coulombicforces exerted by the other nuclei and by a hypothetical electroncloud whose charge density −ρ(r) is found by solving the electronicSchrodinger equation.

This is the Hellmann-Feynman electrostatic theorem.

VHFT

At the equilibrium geometry the effective force on each nucleus iszero, as the derivatives of U respect to all nuclear Cartesiancoordinates are vanishing. Then, the hypothetical electron smearmust exert on each nucleus a net attractive Hellmann-Feynmanforce to counterbalance the internuclear repulsion.

From the Hellmann-Feynman viewpoint, we seem to be consideringchemical bonding solely in terms of potential energy, whereas thevirial-theorem discussion involved both potential and kineticenergy. The use of the electrostatic theorem to explain chemicalbonding has been criticized by some quantum chemists on thegrounds that it hides the role of kinetic energy in bonding.

VHFT

Books

Schiff L. I. “Quantum Mechanics” McGraw-Hill

Levine I. N. “Quantum Chemistry” Prentice-Hall

VHFT