11 Examples of the `Newtonian' Perspective: The Ground and First Excited
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2.11 Examples of the ‘Newtonian’ Perspective: The Ground and First Excited …
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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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2 Schrödinger Theory from the ‘Newtonian’ Perspective …
As the Hamiltonian is both separable and independent of spin, the wavefunction
ψ(x1 x2 ) may be written as
ψ(x1 x2 ) = ψ(r1 r2 )χ(σ1 σ2 ) = φ(s)ξ(R)χ(σ1 σ2 ),
(2.172)
where ψ(r1 r2 ) is the spatial part of the wave function, χ(σ1 σ2 ) the spin component,
and where φ(s), ζ(R) are orbital functions. Since R is symmetric in an interchange
of the spatial electronic coordinates, the function ξ(R) is symmetric. According
to the Pauli exclusion principle then, if the spin function χ(σ1 σ2 ) is symmetric
(triplet state) in an interchange of the electrons, then the orbital function φ(s) must
be antisymmetric [φ(−s) = −φ(s)], and if χ(σ1 σ2 ) is antisymmetric (singlet state),
then φ(s) must be symmetric [φ(−s) = φ(s)]. There are no constraints on the orbital
function ξ(R) due to its symmetry.
The Schrödinger equation Hψ = Eψ then separates into the equations
Hˆ s φ(s) = φ(s),
(2.173)
Hˆ R ξ(R) = ηξ(R),
(2.174)
E = + η.
(2.175)
with the total energy
The normalization condition on ψ also separates into
|φ(s)|2 ds = 1 and
|ξ(R)|2 dR = 1.
(2.176)
The equation for ξ(R), (2.174), is the harmonic oscillator equation whose solutions
are analytical. The reader is referred to the original literature [14] for the solution
of (2.173) for the orbital φ(s). It turns out that closed form analytical solutions exist
only for certain discreet values of the spring constant k. Further, excited states of the
Hooke’s atom are defined in terms of the number of nodes of φ(s). Those solutions
with zero nodes are ground states, those with one node correspond to the first excited
state, and so on. However, the analytical solutions for the ground and excited states
correspond to different values of the spring constant k. The properties of the Hooke’s
atom in a ground and first excited singlet state, and the fields representative of the
different electron correlations, are described in the following subsections [39, 40].
The analytical expressions for these properties are given in Appendix C.
2.11.2 Wavefunction, Orbital Function, and Density
The ground ψ00 (r1 r2 ) and first excited singlet ψ01 (r1 r2 ) state wavefunctions we
consider are
2.11 Examples of the ‘Newtonian’ Perspective: The Ground and First Excited …
ψ00 (r1 r2 ) = ξ0 (R)φ0 (s),
ξ0 (R) =
3/4
2ω
π
φ0 (s) = a00 e−ωs
(2.177)
e−ωR ,
(2.178)
(1 + ωs),
(2.179)
2
2
/4
47
√
√
√
where√ a00 = ω 5/4 (3π π/2 + 8π ω + 2π 2πω)−1/2 = 1/14.55670, k = 1/4,
ω = k = 1/2;
(2.180)
ψ01 (r1 r2 ) = ξ0 (R)φ1 (s)
φ1 (s) = a01 e−ωs
2
/4
1 + C1
ω
ω
ω
s + C2 s2 + C3
2
2
2
3/2
s3 ,
(2.181)
√
√
where a01 = ω 3/4 [8 2π(C1 + 2C1 C2 + 2C3 + 6C2 C3 ) + π 2π( 15
C 2 + 105
C32 +
2 2
4
2
−1/2
= 1/13.21931,
C1 = 1.146884, C2 = −0.561569,
3C1 + 6C2 + 15C1 C3 + 2)]
√
C3 = −0.489647, k = 0.144498, ω = k = 0.380129.
In order to provide a pictorial representation [41] of the wave function and to
exhibit the electron-electron coalescence in its structure, we plot in Fig. 2.1 the ground
state wave function ψ00 (r1 r2 ) of (2.177) for θr1 ,r2 = 0◦ , where θr1 ,r2 is the angle
Fig. 2.1 Structure of the ground state wave function ψ00 (r1 r2 ) for θr1 ,r2 = 0◦ , where θr1 ,r2 is the
angle between vectors r1 and r2 which are oriented along both the positive and negative z-axis
48
2 Schrödinger Theory from the ‘Newtonian’ Perspective …
Fig. 2.2 Same as in Fig. 2.1 but for θr1 ,r2 = 90◦ . The vector r1 is along the z-axis, and the vector
r2 in the xy-plane
between the vectors r1 and r2 . Figure 2.2 is a plot for θr1 ,r2 = 90◦ . Figure 2.3 is the
same as Fig. 2.2 except that the r2 vector has been confined to the positive quadrant
of the xy-plane. Observe that the electron-electron coalescence cusp is clearly visible
along the diagonal defined by r1 = r2 in Fig. 2.1, and at r1 = r2 = 0 at the nucleus
in Fig. 2.3.
In both the ground and first excited singlet state, the atom is spherically symmetric.
The orbital functions φ0 (s) and φ1 (s) are plotted in Fig. 2.4. Note that there are no
nodes in φ0 (s), and one node in φ1 (s) corresponding to a first excited state. Also
observe the electron-electron coalescence cusp at the coalescence of the two electrons
for s = 0. In Fig. 2.5 the ground ρ00 (r) and excited ρ01 (r) state densities are plotted.
Recall that this source is static in that its structure is independent of and remains
unchanged as a function of electron position. Since the potential energy v(r) is not
Coulombic, these densitites do not exhibit a cusp at the nucleus. The corresponding
radial probability densities r 2 ρ00 (r) and r 2 ρ01 (r) are plotted in Fig. 2.6. Observe
the distinct shoulder in r 2 ρ01 (r) prior to the maximum indicative of a ‘shell’ type
structure with each electron being in a different shell.
2.11 Examples of the ‘Newtonian’ Perspective: The Ground and First Excited …
49
Fig. 2.3 Same as in Fig. 2.2 except that the r2 vector has been confined to the positive quadrant of
the xy-plane
Fig. 2.4 The relative
coordinate component of the
wavefunction for the ground
φ0 (s) and the first excited
singlet φ1 (s) states
50
Fig. 2.5 The electron
density ρ(r) of the ground
and first excited singlet states
Fig. 2.6 The radial
probability density r 2 ρ(r) of
the ground and first excited
singlet states. The arrows
indicate the position of the
maxima
2 Schrödinger Theory from the ‘Newtonian’ Perspective …
2.11 Examples of the ‘Newtonian’ Perspective: The Ground and First Excited …
51
2.11.3 Fermi–Coulomb Hole Charge Distribution ρxc (rr )
In Figs. 2.7, 2.8, 2.9 and 2.10, the Fermi–Coulomb hole charge distribution ρxc (rr )
for both the ground and excited states is plotted for electron positions at the nucleus
r = 0, and at r = 0.5, 1, 2, 10, 20, 50, and 200 a.u. The electron position is indicated by an arrow. For the electron at the nucleus Fig. 2.7a, the hole is spherically
symmetric about the electron. Further, at the electron position, the hole exhibits
a cusp representative of the electron–electron coalescence condition of (2.149). In
Figs. 2.7b, 2.8, 2.9 and 2.10, the electron is along the z-axis corresponding to θ = 0◦ .
The cross sections plotted correspond to θ = 0◦ with respect to the electron–nucleus
direction. The graph for r < 0 is the structure for θ = π and r > 0.
The dynamic or nonlocal nature of the Fermi–Coulomb hole as a function of
electron position is clearly evident in these figures, as is the cusp at the electron
position in Figs. 2.7b, 2.8a, 2.9b, and the fact that these holes are not spherically
symmetric about the electron. For asymptotic positions of the electron, these charge
distributions become essentially spherically symmetric about the nucleus as well as
static (Fig. 2.10b). In other words, the change in the structure for these asymptotic
positions is minimal. Finally, observe that the structure of the holes for the ground
and excited states is distinctly different, although their broad features are similar.
Fig. 2.7 Cross–section through the Fermi–Coulomb hole charge ρxc (rr ) for the ground and first
excited singlet states. In (a) the electron is at the nucleus r = 0, and in (b) at r = 0.5 a.u. The
electron is on the z-axis corresponding to θ = 0. The graphs for r < 0 correspond to the structure
for θ = π, r > 0
52
2 Schrödinger Theory from the ‘Newtonian’ Perspective …
(a)
(b)
Fig. 2.8 Same as in Fig. 2.7 but for the electron at (a) r = 1 a.u. and (b) r = 2 a.u
(a)
(b)
Fig. 2.9 Same as in Fig. 2.7 but for the electron at (a) r = 10 a.u. and (b) r = 20 a.u
2.11 Examples of the ‘Newtonian’ Perspective: The Ground and First Excited …
(a)
53
(b)
Fig. 2.10 Same as in Fig. 2.7 but for the electron at (a) r = 50 a.u. and (b) r = 200 a.u
2.11.4 Hartree, Pauli–Coulomb, and Electron–Interaction
Fields E H (r), E xc (r), E ee (r) and Energies
EH , Exc , Eee
The Hartree field E H (r) whose source is the density ρ(r) (see (2.47)) is plotted
in Fig. 2.11 for the ground and excited states. Since for each state, the density is
spherically symmetric, the field vanishes at the nucleus. The fact that there is a single
‘shell’ is evident from the ground state plot. A careful examination of the field for
the excited state shows a slight shoulder between r = 2 and 4 a.u. indicating the
existence of the second ‘shell’. As the density is static, localized about the nucleus,
and of total charge 2 a.u., the structure of the Hartree field E H (r) for asymptotic
positions of the electron is
E H (r) =
ρ(r )(r − r )
dr
|r − r |3
r→∞
1
r2
ρ(r )dr =
2
.
r2
(2.182)
The fields E H (r) for both the ground and excited state are observed to merge asymptotically with the function 2/r 2 also plotted in Fig. 2.11.
The Pauli–Coulomb field E xc (r) for both states is plotted in Fig. 2.12. Since for
the electron position at the nucleus, the Fermi–Coulomb hole charge ρxc (rr ) is
spherically symmetric about the electron (see Fig. 2.7a), the fields E xc (r) vanish
there. Further, as the atom is spherically symmetric, the field E xc (r) has only a radial
component and is dependent only on the radial coordinate. This is the case in spite of
the fact that the Fermi–Coulomb hole is not spherically symmetric about the nucleus
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Fig. 2.11 The Hartree field
E H (r) for the ground and
excited states
Fig. 2.12 The
Pauli-Coulomb field E xc (r)
for the ground and excited
states
2 Schrödinger Theory from the ‘Newtonian’ Perspective …
2.11 Examples of the ‘Newtonian’ Perspective: The Ground and First Excited …
55
for other electron positions. The fields are negative because the Fermi–Coulomb
hole charge is negative. The two ‘shells’ are clearly evident in the field E xc (r) for
the excited state, and the single shell for the ground state. As noted previously, for
asymptotic positions of the electron, the Fermi–Coulomb hole is essentially a static
charge and spherically symmetric, and localized about the nucleus. Since the total
charge of the hole is negative unity, the asymptotic structure of the Pauli–Coulomb
field is
1
(2.183)
E xc (r) ∼ − 2 .
r→∞
r
Once again, the field E xc (r) for both the ground and excited state merge asymptotically with the function −1/r 2 also plotted in Fig. 2.12. This result is general and
valid for any finite system.
The electron–interaction field E ee (r) which is the sum of the Hartree E H (r) and
Pauli–Coulomb E xc (r) fields is plotted in Fig. 2.13 for both states. Since the total
charge of its source, the pair-correlation density, is unity, the fields decays as 1/r 2
asymptotically. For purposes of comparison with the other components of the internal
field F int (r) experienced by the electrons, E ee (r) is also plotted in Fig. 2.16 for the
ground state and in Fig. 2.17 for the excited state.
The Hartree EH , Pauli–Coulomb Exc , and electron–interaction Eee energies as
determined from the corresponding fields are given in Table 2.1. A comparison of
the fields for the ground and excited states makes clear why the Hartree and Pauli–
Coulomb energies for the former are greater in magnitude. The graphs of the fields
also show the region of space from which the principal contribution to the energy
arises.
Fig. 2.13 The
electron–interaction field
E ee (r) for the ground and
excited states