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
4π
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)