5.73 Lecture #29
29 - 1
Begin Many-e– Atoms: Quantum Defect Theory
See MQDT Primer by Stephen Ross, pages 73-110 in Half Collision
Resonance Phenomena in Molecules (AIP Conf. Proc. #225,
M. Garciá-Sucre, G. Raseev, and S.C. Ross) 1991.
Last Time:
e2
+
* turning points of V (r ) = −
r
2
( + 1)
r2
h l l
l
  l(l + 1)  1/ 2 
l(l + 1) 
2
r± (n, l ) = a0 n 1 ± 1 −
a
n
1
1
m
for l << n
≈
±

0

n2  
2 n 2 
 
2
* µn (r)≡rRn (r) dominated by small lobe (n-independent nodal position) at
inner turning point, amplitude scales as n–3/2, and large lobe at outer
l
envelope
un (r ) ∝ pn (r )−1/ 2
l
l
l
turnin point (essentially all of the probability.
ℜ
*
En = IP −
*
nodes:
*
expectation value scaling
l
n2
n − l − 1 radial nodes
l angular nodes
n − 1 total nodes
λ 2 gives spacing between radial nodes
r σ σ < −1
∝ n −3
σ ≥ +1
∝ n 2σ
(see below)
σ = −1
∝ n −2
(H - atom energy levels)
geometric mean of expectation values of r for off - diagonal matrix elements
r
nl
[
∝ n2
∝ ( r+ nl )
1/2
] ≈ [(n )
1/2 σ
( r+ n ′l ′ )
( when is r n ≈ r
2 1/2
( n ′ 2 )1/2 ]
σ
σ
= ( nn ′ ) ≈ n 2σ
n
?)
updated September 19,
5.73 Lecture #29
29 - 2
TODAY
1. Many-e– atom treated as core plus outer e– that sees shielded core as Z(r).
2.
l
-dependent energy shifts → n-independent quantum defects E nl = IP −
ℜ
( n − µ nl )2
ν
3. energy shifts are actually phase shifts in unl(r) relative to unl(r) for H-atom
4. Rigorous QDT
A. regular and irregular Coulomb functions f,g satisfy Hydrogen-like Schr. Eq.
OUTSIDE core
B. Boundary conditions at r → ∞
1/2
 ℜ 
noninteger values of ν =  −
 require mixture of f and g
 E nl 
find ν = n − µ l satisfies r → ∞ boundary condition
∞ number of members in series of ν©s with integer spacings, ∴ constant quantum defect
C. πµl is a phase shift
repeated patterns in each integer region of ν
D. Multi-channel QDT
µ matrices
e– colliding with core can also transfer energy and angular
momentum to core-e–
* channels rather than eigenstates
* focus on dynamics, but in a “black box” way. Dynamics happens within a
restricted region of space. This region of space is always sampled, regardless of
E, in the same way. Everything is determined by the boundary conditions for
the outgoing wave.
SCATTERING THEORYrather than EFFECTIVE HAMILTONIAN MODEL.
The goal here is to extract from a complicated many-body problem some regular
features that will help in assigning and modeling experimental data.
updated September 19,
5.73 Lecture #29
29 - 3
1. Many-e– Atom
e–
r0
•
outer electron
(valence, Rydberg)
+Z
Z
–
outside core e sees Z = +1
inside core e– sees Z(r)
Shell Structure
1
ensures n–3
rather than n–2
rcore
2.
l
as r → 0, H(1) diverges faster than − e 2 r
because Z(0) > 1.
-dependent sampling of core
high l : see Z (r ) ~ +1
low l : see Z (r ) = Z eff >> 1
−
l
energy stabilization
e2
r
⇓
H=H
H(1) ≈ −
∴ E (n1) = nl H(1) nl ∝ −
l
( 0)
[ Z(r ) − 1]e 2
⇓ ( )
1
l
l
l
∴ µ =
c
l
2ℜ
) expand in Taylor series
2
l
+H
r
ℜ c
−ℜ
∴ En = − 2 − 3 ≈
n
n
(n − µ
c
n3
l
<< n
call this ν, effective
principal qn
so far we have focussed on Enl
3. What does Z(r) do to unl(r)?
* outside core sees same Vl ( r ) as H
* must be same as u nl ( r ) for H except for phase shift inward (why inward?)
* all the unique stuff is inside core – causes the phase shift.
- nodal structure inside core is invariant wrt n or E
updated September 19,
5.73 Lecture #29
29 - 4
Mulliken: “ontology recapitulates phylogeny”
intra-core nodal structure is n-independent
nodal structure encodes all e–↔nucleus dynamics!
4. Do all of this more rigorously: QDT
*
*
*
*
regular Rydberg series, one for each l
These are what
n-scaling of inner lobe amplitude and of all matrix elements we will obtain.
large quantum defects for small l
entire Rydberg series and associated ionization continuum (e– ejected
in l-partial wave) is a single entity
follow Ross but not using atomic units
redefine 0 of E
En = −
ℜ
n2
n = 1, 2,… for H
E
n = − n 
 ℜ
−1 / 2
−1/2
 E 
ν ≡ − ν 
 ℜ
and use ν (™effective principal quantum number) rather than E as a label foru nl ( r )
generalize to noninteger n for non - hydrogen:
Schrödinger Equation for H (the “Coulomb Equation”)
 h  d
−
 2−
 2µ  dr
2
2
(

+ 1)  e
 − − E u (ν , r ) = 0
2

r
 r
l l
as variable rather
than as q.n.
2
l
E=−
ℜ
ν2
well known solutions:
2nd order differential equation - two linearly independent
solutions (at each l,ν)
fl ( ν, r ) → 0 as r → 0
g l ( ν, r ) → ∞ as r → 0
" regular"
" irregular"
of no use for Hydrogen, but it turns out that we need
both f and g to satisfy r → ∞ boundary condition.
updated September 19,
5.73 Lecture #29
29 - 5
for H, we have no use for gl(ν,r) because it cannot satisfy boundary
conditions as r → 0
A. For many-e– atoms, beyond some critical r0, Schr. Eq. is identical to that of
H – the only difference is that we must treat the r → 0 boundary condition
differently
Universal boundary conditions are
r → ∞, u l ( ν, r ) → 0
for E < 0, r → ∞, asymptotic forms for f and g are
f ( ν, r → ∞) → C(r ) sin[ π( ν − l )]e r / ν
l
g ( ν, r → ∞) → − C(r ) cos[ π( ν − l )]e r / ν
l
C(r ) → 0 as r → ∞
but C(r )e r / ν → ∞ as r → ∞, so the only way to satisfy the r → ∞ boundary
condition for a pure u ( ν, r ) = f ( ν, r ) is for ( ν − l ) = integer
l
l
B. But we might want to use a mixture of fl and gl to deal with non-integer
(ν – l), as we will need for many-electron atoms.
H
Na
Z = +11
Na ul(ν,r) emerges from core with
extra phase – sucking in of hydrogenic
function
core
* invariance of intra-core nodal structure – amount of
phase shift should be independent of n. [We expect
this to be true.]
u (ν , r) = αf (ν , r ) − (1 − α 2 )
1/ 2
l
l
g (ν , r)
l
**
* mixing of 2 types of function is required in order to
have noninteger ν, yet still satisfy ul(ν,r)→0 as r →
∞ boundary condition
updated September 19,
5.73 Lecture #29
29 - 6
α ≡ cos( πµ l )
TRICK:
−(1 − α 2 )
1/2
 plug this into * * equation

= − sin ( πµ l )  on page 29 - 5
[
(
)
(
ψ = u (ν , r)Φ (CORE) = f (ν , r) cos πµ − g (ν , r) sin πµ
14243
l
l
l
l
l
l
)]Φ
l
other e −
plug in asymptoptic forms for f,g
{
ψ ⇒ sin[ π(ν − l )] cos πµ + cos[ π(ν − l )] sin πµ
(
l
)
f (ν , r)
–g (ν,r)
{ } → 0 as r → ∞ is required. How?
l
{
(
)}C (r )e
r/ ν
l
Φ
l
l
} = 0 : sin[ π( ν − l)] cos( πµ ) = − cos[ π( ν − l)]sin( πµ )
l
sin( πµ
sin[ π( ν − l )]
=−
cos[ π( ν − l )]
cos( πµ
l
)
)
)
l
tan[ π( ν − l )] = − tan( πµ
l
constraint on ν. What are all of
the values of ν which are
consistent with this constraint?
l
tan θ = − tan( −θ) = − tan( −θ + n′π)
let θ = πµ
↑
integer
l
∴ tan[ π( ν − l )] = tan(– θ + n′π)
thus − θ + n ′π = π( ν − l)
but πµ = π( n ′ − ν + l )
⇒
θ = n ′π − π( ν − l)
θ = π( n ′ − ν + l)
n = n ′ + l, µ l = n − ν
l
n′ + l ≡ n
↑
integer
ν = n – µl
ν is smaller than integer n
by constant term µl.
Implies existence of infinite series of values of ν for which ψ → 0 as r → ∞.
updated September 19,
5.73 Lecture #29
29 - 7
Get this infinite series of ν’s, increasing in steps of 1, simply by
specifying one ν-independent value of µl!
All of the ν-dependence (E-dependence) of ψl(ν,r) is explicitly built
into fl(ν,r) and gl(ν,r). µl describes the relative amounts of fl and gl
in ψ. This f,g mixing is determined when the e– leaves the core
with the precise phase shift so that ψ → 0 at r → ∞.
C. How can we show that πµl is a phase shift?
The asymptotic form of ψ is
{
ψ → sin[ π( ν − l )] cos( πµ ) + cos[ π( ν − l )]sin( πµ
l
l
)}C(r )er / ν
use double angle formula sin A cos B + sin B cos A = sin( A + B)
{ [
ψ → sin π( ν − l ) + πµ
l
]}
C(r )e r / ν
but f ( ν, r ) → sin[ π( ν − l )]C(r )e r / ν
l
so this modified function is identical to the regular [i.e. fl(ν,r)]
Coulomb function but with a πµl phase shift.
If µl > 0, this corresponds to an advance of the phase of ul(ν,r)
relative to that for H. As expected, ψ is sucked into core by an
amount πµl [+ an arbitrary number of 2π’s] by the Z(r) core.
πµl is the phase shift that occurs inside the core. It is the
boundary condition at r = 0 shifted out to r = r0. On the other
hand, the r → ∞ b.c. is satisfied by ν = n – µl where n is integer.
the
quantum
defect!
updated September 19,
5.73 Lecture #29
29 - 8
exact replica
of lower-n
pattern
n+1
n
s
p
d
f
…
µ s >> µ p > µ d > µ f … ≈ 0
everything is repeated in each integer region of ν
ν, not E, is the way to look at Rydberg “patterns”
Finding the way to see a pattern is ALWAYS the route to both
“assignment” and “insight”
µ l decreases as l increases because of the expected behavior of
Z eff ( r ) as sampled in the presence of a centrifugal barrier
∝
( + 1)
l l
2µr 2
updated September 19,
5.73 Lecture #29
29 - 9
D. inter-series interactions? Suppose you have B 2s22p1
B + (2 s 2 p)
autoionization
B + (2 s 2 )
perturbation
B 2s2 2 p 2P
Separate series converging to 2 series limits
perturbations
autoionization
updated September 19,
5.73 Lecture #29
29 - 10
Described by a Multichannel Quantum Defect Theory
µ s, µ p , µ d
Replace
etc.
by 3 × 3 µ matrices, one for each symmetry
[2s ( S) ⊗ ν ]
[2s2 p( P ) ⊗ ν
[2s2 p( P ) ⊗ ν
2 1
1l
3
1
2l
3l
2
]
± 1]
±1
2
2
l
l
l





more complicated
many-electron
coupling problem
subject of next few
lectures.
Overall symmetry: H is totally symmetric.
off-diagonal elements describe inter-channel interactions (exchange of
angular momentum between Rydberg e– and core e–s.)
describe what happens in a collision of e– with ion-core. Does it change
the state of the ion? Does it change the kinetic energy and/or angular
momentum of the e–? Unified picture of scattering at negative E (bound
states) and at positive E.
Next few lectures:
states of many-electron atoms
How to calculate matrix elements of many-electron (many Fermion)
systems.
updated September 19,