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10 Time-Independent Schrödinger Theory: Ground and Bound Excited States

10 Time-Independent Schrödinger Theory: Ground and Bound Excited States

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2.10 Time-Independent Schrödinger Theory: Ground and Bound Excited States



39



The fields E ee (r), D(r), and Z(r) are representative of correlations between the

electrons due to the Pauli exclusion principle and Coulomb repulsion, the density,

and kinetic effects, respectively. Since, by assumption, the external field F ext (r) is

conservative (∇ × ∇v(r) = 0), so is the internal field F int (r).

Again, the external potential energy v(r) can be afforded a rigorous physical

interpretation via the ‘Quantal Newtonian’ first law: It is work done to move an

electron from some reference point at infinity to its position at r in the force of the

conservative internal field F int (r):

v(r) =



r





∇v(r ) · d



=



r





F int (r ) · d .



(2.136)



The work done is path-independent.

Since the internal field F int (r) is obtained from quantal sources that are expectations of Hermitian operators taken with respect to the eigenfunctions ψn (X), the

potential energy v(r) is a functional of these eigenfunctions: v(r) = v[ψn ]. Thus,

the time-independent Schrödinger equation (2.133) may be written as





1

2



∇i2 +

i



1

2



i,j



1

+

|ri − rj |



vi [ψn ] ψn = En ψn ,



(2.137)



i



where vi = v(ri ). This demonstrates the self-consistent nature of the Schrödinger

equation. Written more explicitly in terms of the internal field F int (r) we have

(2.137) to be





1

2



∇i2 +

i



1

2



i,j



1

+

|ri − rj |



ri

i







F int (r) · d



ψn = En ψn . (2.138)



In order to solve the Schrödinger equation, one begins with an approximation to ψn .

With this wave function one obtains the quantal sources and thereby the internal field

F int (r), and solves the integro-differential equation (2.138) to obtain the new solution

ψn and eigenvalue En . This process is continued till self-consistency is achieved, and

the exact ψn , En obtained.

The integral virial theorem is the time-independent version of (2.82):

ρ(r)r · F ext (r)dr + Eee + 2T = 0.



(2.139)



Finally, the average and averaged torque of the internal field F int (r) vanishes:

ρ(r)F int (r)dr = 0,



(2.140)



40



2 Schrưdinger Theory from the ‘Newtonian’ Perspective …



ρ(r)r × F int (r)dr = 0,



(2.141)



since the contribution of each component vanishes.

To reiterate, the perspective of time-independent Schrödinger theory in terms of

fields and quantal sources representative of the different electron correlations, is valid

for both ground and bound excited pure states whether non-degenerate or degenerate.

In Sect. 2.11 this perspective is described for a ground and excited state of the Hooke’s

atom. In addition, the perspective brings out the intrinsic self-consistent nature of the

Schrödinger equation. The self-consistent form of the Schrödinger equation (2.138)

also makes clear that for different self-consistently obtained solutions ψn , there exist

different external potentials v(r).



2.10.2 Coalescence Constraints

As a consequence of the Coulomb interaction, the Hamiltonian (2.131) is singular

when two electrons coalesce. It is also singular for the case where the potential energy

v(r) is Coulombic as when an electron coalesces with the nucleus of charge Z. In

order for the wavefunction ψ(X) to satisfy the Schrödinger equation (2.133) and

remain bounded, it must satisfy a coalescence condition at each singularity. These

coalescence constraints play a significant role in Q-DFT and other local effective

potential energy theories as discussed later in the section. There are two forms of

these coalescence constraints: the integral and differential forms. The integral form

is more general in that it retains the angular dependence of the wave function at

coalescence, and the differential form can be readily derived from it. Historically, it

was the differential form that was originally derived [27], and we follow that path of

description in this section.

With s = r − r , and r, r the positions of the two particles, the differential form

of the coalescence condition on the wavefunction is

dψsp.av

|s=0 = ζψ|s=0 ,

ds



(2.142)



where ψsp.av is the spherical average of the wavefunction about the singularity:

ψsp.av (s) =



1





ψd



s.



(2.143)



For the electron–electron cusp condition, the coeffiecient ζ = 21 ; for the electron–

nucleus cusp condition ζ = −Z.

The electron–nucleus coalescence condition may also be expressed [28] in terms

of the derivative of the density and density at the nucleus. Thus, with the timeindependent density defined as (see (2.11))



2.10 Time-Independent Schrödinger Theory: Ground and Bound Excited States



ψ ∗ rσ, XN−1 ψ rσ, XN−1 dXN−1 ,



ρ(r) = N



41



(2.144)



σ



we have on taking the derivative in the limit of the electron–nucleus coalescence

dρ(r)

=N

r→0 dr



dψ ∗ (rσ, XN−1 )

dr



lim



σ



+ ψ ∗ (r = 0, σ; XN−1 )



r→0



ψ r = 0, σ; XN−1



dψ(rσ, XN−1 )

dr



r→0



dXN−1 .



(2.145)



Integrating the previous equation over the angular variables of the coalescing electron

we obtain

dρ(r)

=N

r→0 dr





dψsp.av

(rσ, XN−1 )



lim



dr



σ



+ ψ ∗ (r = 0, σ; XN−1



r→0



dψsp.av (rσ, XN−1 )

dr



ψ r = 0, σ; XN−1



r→0



dXN−1 ,



(2.146)



which on substituting the cusp condition on the right hand side leads to

ψ ∗ (r = 0, σ; XN−1 )ψ(r = 0, σ; XN−1 )dXN−1 .



= −2ZN



(2.147)



σ



The electron-nucleus coalescence or cusp condition in terms of the density is then

lim



r→0



dρ(r)

= −2Zρ(r = 0).

dr



(2.148)



Thus, the densities in atoms, molecules, and solids exhibit a cusp at the nuclei. The

cusp for electron–electron coalescence is exhibited in the structure of the Fermi–

Coulomb hole charge distribution.

The integral form of the cusp coalescence constraint for an arbitrary state of a

system of N charged particles as particles 1 and 2 coalesce is

ψ(r1 , r2 , . . . rN ) = ψ(r2 , r2 , r3 , . . . rN )(1 + ζr12 )

2

+ r12 · C(r2 , r3 , . . . rN ) + O(r12

).



(2.149)



Here r12 = |r1 − r2 |, r12 = r1 − r2 , and C(r2 , r3 , . . . rN ) an undetermined vector.

The spin index is suppressed. The integral form of the coalescence condition was

originally [29] a conjecture. It can, however be derived [30, 31] directly from the

Schrödinger equation. The integral form of the coalescence condition retains the

angular dependence of the wave function at coalescence, and is thus more general and

useful. The differential form of the coalescence condition (2.142) is readily obtained



42



2 Schrödinger Theory from the ‘Newtonian’ Perspective …



by taking the spherical average and differentiating about the point of coalescence. It

is evident from the integral cusp coalescence condition (2.149) and the definition of

the density (2.144), that there can be no differential form similar to (2.148) in terms

of the density of the cusp condition for electron-electron coalescence. However, it is

possible to derive the integral and differential forms of the coalescence constraints

for the time-independent pair function P(rr ) of (2.27) (see [32] and Chap. 2 of

QDFT2). Note that the integral coalescence expression is equally valid even if the

wave function vanishes at the point of coalescence, i.e. if ψ(r2 , r2 , . . . , rN ) = 0. This

is referred to as a node coalescence condition as opposed to the cusp coalescence

condition.

Employing the integral form of the electron-nucleus coalescence constraint, it can

be proved [33] (see also Chap. 8 of QDFT2) that the local effective potential energy

function within Q-DFT which incorporates all the many-body effects is finite at the

nucleus. This is also the case for all other local effective potential theories. (Prior to

[33–35], there was controversy in the literature with regard to the structure of the

potential at and near the nucleus. For a brief historical description of this controversy,

and for the derivation of this structure, see Chap. 8 of QDFT2.)

For the generalization of the derivation of the integral coalescence condition to

dimensions D ≥ 2 see [32] and Chap. 2 of QDFT2



2.10.3 Asymptotic Structure of Wavefunction and Density

Another important property of the wavefunction and density is their asymptotic

structure in the classically forbidden region because this structure is related to the

first ionization potential. (This fact is significant in providing a rigorous physical

interpretation of the highest occupied eigenvalue within Q-DFT (see Sect. 3.4.8) and

other local effective potential energy theories.) To show this [36, 37] we rewrite the

N–electron Hamiltonian of (2.131) as

1

Hˆ = − ∇ 2 + v(r) +

2



N



i=2



1

+ Hˆ N−1 ,

|r − ri |



(2.150)



where the (N − 1) electron Hamiltonian Hˆ N−1 is

1

Hˆ (N−1) = −

2



N



N



∇i2 +

i=2



N



v(ri ) +

i=2



1

1

.

2 i=j=1 |ri − rj |



(2.151)



Now the complete set of eigenfunctions and eigenenergies of the (N − 1)–electron

system are defined by the equation

Hˆ N−1 ψs(N−1) (XN−1 ) = Es(N−1) ψs(N−1) (XN−1 ).



(2.152)



2.10 Time-Independent Schrödinger Theory: Ground and Bound Excited States



43



Next expand the N–electron wavefunction ψn (X) (see 2.133) in terms of the eigenfunctions ψs(N−1) :

Csσ (r)ψs(N−1) (XN−1 ),



ψn (rσ, XN−1 ) =



(2.153)



s



and rewrite the Schrödinger equation (2.133) as

1

− ∇ 2 + v(r) +

2



N



i=2



1

+ Hˆ (N−1)

|r − ri |



Csσ (r)ψs(N−1) (XN−1 )

s



Csσ (r)ψs(N−1) (XN−1 ).



= En



(2.154)



s



For asymptotic positions of the electron we have by Taylor expansion

1 ri · r 1

1

= + 3 +

|r − ri |

r

r

2



riα riβ

α,β



∂2 1

+ ...,

∂rα ∂rβ r



(2.155)



so that on retaining just the leading term, (2.154) becomes

1

N −1

− ∇ 2 + v(r) +

2

r

En −



=



Csσ (r)ψs(N−1) (XN−1 )

s



Es(N−1)



Csσ (r)ψs(N−1) (XN−1 ).



(2.156)



s



Multiplying (2.156) by ψs(N−1)∗ (XN−1 ) from the left, integrating over

employing the orthonormality condition



we have



dXN−1 , and



ψs(N−1) |ψs(N−1) = δss ,



(2.157)



1

N −1

− ∇ 2 + v(r) +

+ Is,n Csσ (r) = 0,

2

r



(2.158)



where the ionization potential Is,n is

Is,n = Es(N−1) − En .



(2.159)



The Is,n are the ionization potentials from the N–electron state with energy En into

various states of the (N − 1)–electron ion. It is assumed that Is,n < Is+1,n etc.

For atomic systems, v(r) = −Z/r. For molecules in the far asymptotic region,

v(r) = −Q/r, where Q is the total nuclear charge. Thus, the Schrödinger equation



44



2 Schrödinger Theory from the ‘Newtonian’ Perspective …



in the asymptotic region is of the form

1

(Z − N + 1)

− ∇2 −

+ Is,n Csσ (r) = 0,

2

r



(2.160)



and the asymptotic solution is

Csσ (r)r→∞ r βs e−αs r χ(σ),



(2.161)



where (1 + βs ) = (Z − N + 1)/αs , and αs = 2Is,n . The satisfaction of the differential equation (2.160) with this solution occurs on neglecting the 0(1/r 2 ) term of

∇ 2 Csσ (r).

The density ρ(r) defined by (2.144) employing (2.153) is then

|Csσ (r)|2 ,



ρ(r) = N

σ



(2.162)



s



so that asymptotically

ρ(r)r→∞ exp (−2αs r) = exp −2 2Is,n r .



(2.163)



Thus, the asymptotic structure of the density is related to the first ionization potential Is,n . This is the case whether the system is in a ground or excited state. For

asymptotic positions of the electron in finite systems, it has been shown [38] that if

the (N − 1)–electron ion ground state is degenerate, then the eigenfunctions ψs(N−1)

and hence the ground–state wavefunction ψ0 , depend parametrically on the direction

of electron removal. This then translates to a parametric dependence on this direction for the asymptotic structure of the single particle density matrix γ(rr ) and the

pair-correlation density g(rr ) [38].

For the derivation of the asymptotic structure to higher order of the wave function ψn (X), density ρ(r), single-particle density matrix γ(rr ), and pair-correlation

density g(rr ), see Chap. 7 of QDFT2



2.11 Examples of the ‘Newtonian’ Perspective: The Ground

and First Excited Singlet State of the Hooke’s Atom

2.11.1 The Hooke’s Atom

The physics underlying the ‘Newtonian’ perspective of Schrödinger theory is demonstrated in this section by application to the analytically solvable Hooke’s atom [14]

in both its ground and first excited singlet state. This atom comprises of two electrons

in an external field such that the potential energy v(rt) due to the field is of the form



2.11 Examples of the ‘Newtonian’ Perspective: The Ground and First Excited …



45



v(rt) = v0 (r) for t ≤ t0

= v0 (r) + v1 (rt) for t > t0 ,



(2.164)



where v0 (r) = 21 kr 2 , k is the spring constant, v1 (rt) = −F(t) · r, with the force

F(t) arbitrary. The Coulomb interaction between the electrons is treated exactly

in this model atom. Based on the Harmonic Potential Theorem of Sect. 2.9, the

wavefunction for t > t0 is the time-independent solution for t ≤ t0 , multiplied by

a phase factor, and shifted by the function y(t) satisfying (2.129). Thus, the time

evolution of all observables is known exactly for t > t0 . However, for properties

that are the expectation value of Hermitian operators such as the density, the time

evolution is the same as that of the property derived for t ≤ t0 but translated by a

finite time-dependent value. Hence, we describe here a study via the ‘Newtonian’

perspective of the system in its stationary state.

The time-independent Hamiltonian for the Hooke’s atom is

1

1

1

1

1

,

Hˆ = − ∇r21 − ∇r22 + kr12 + kr22 +

2

2

2

2

|r1 − r2 |



(2.165)



where r1 and r2 are the coordinates of the electrons. This Hamiltonian is separable

by transforming to the relative and center of mass coordinates:



so that



and

∇r21 =



R=



s

r1 = R + ;

2



s

r2 = R − ;

2



1 2

1

∇ + ∇s2 + ∇R · ∇ s ; ∇r22 = ∇R2 + ∇s2 − ∇ R · ∇ s .

4 R

4



The Hamiltonian is then



where



r1 + r2

2



s = r1 − r2 ;



(2.166)



(2.167)



(2.168)



Hˆ = Hˆ s + Hˆ R



(2.169)



1

1

Hˆ s = −∇s2 + ks2 + ,

4

s



(2.170)



1

Hˆ R = − ∇R2 + kR2 .

4



(2.171)



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