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Universita’ dell’Insubria, Como, Italy
Boundary-condition-determined wave
functions (and their nodal structure) for
few-electron atomic systems
Dario Bressanini
http://scienze-como.uninsubria.it/bressanini
Critical stability V (Erice) 2008
Numbers and insight
There is no shortage of accurate calculations for
few-electron systems
−2.90372437703411959831115924519440444669690537 a.u.
Helium atom (Nakashima and Nakatsuji JCP 2007)
However…
“The more accurate the calculations became, the
more the concepts tended to vanish into thin air “
(Robert Mulliken)
2
The curse of YT
Currently Quantum Monte Carlo (and quantum
chemistry in general) uses moderatly large to
extremely large expansions for Y
Can we ask for both accurate and compact wave
functions?
3
VMC: Variational Monte Carlo
Use the Variational Principle
H
Y (R ) HY (R )dR
E
Y (R)dR
2
0
Use Monte Carlo to estimate the integrals
Complete freedom in the choice of the trial wave function
Can use interparticle distances into Y
But It depends critically on our skill to invent a good Y
4
QMC: Quantum Monte Carlo
Analogy with diffusion equation
Wave functions for fermions have nodes
If we knew the exact nodes of Y, we could exactly
simulate the system by QMC
The exact nodes are unknown. Use approximate
nodes from a trial Y as boundary conditions
+
-
5
Long term motivations
In QMC we only need the zeros of the wave
function, not what is in between!
A stochastic process of diffusing points is set up
using the nodes as boundary conditions
The exact wave function (for that boundary
conditions) is sampled
We need ways to build good approximate nodes
We need to study their mathematical properties
(poorly understood)
6
Convergence to the exact Y
We must include the correct analytical structure
Cusps:
r12
Y (r12 0) 1
2
Y(r 0) 1 Zr
QMC OK
3-body coalescence and logarithmic terms:
Tails and fragments:
QMC OK
Usually neglected
7
Asymptotic behavior of Y
Example with 2-e atoms
1 2
1 1
1
2
H (1 2 ) Z ( )
2
r1 r2 r12
1 2
Z Z 1
2
H (1 2 )
2
r1
r2
r2
r2
Y 0 (r1 )r2
0 (r1 )
( Z 1) / 1 r2
e
2 EI
is the solution of the 1 electron problem
8
Asymptotic behavior of Y
The usual form
Y (r1 ) (r2 )
Ye
a ( r1 r2 )
does not satisfy the asymptotic conditions
Y (r2 ) 0 (r1 ) (r2 )
Y (r1 ) (r1 ) 0 (r2 )
A closed shell determinant has the wrong structure
Y (r1 ) (r2 ) (r2 ) (r1 )
9
Asymptotic behavior of Y
r1
In general Y0N r1a1 (1 c1r11 O(r1 2 )) e r1 / b1 Yl1m1 (r1 )Y0N 1 (2,... N )
Recursively, fixing the cusps, and setting the right symmetry…
U
ˆ
Y A( f1 (1) f 2 (2)... f N ( N )N )e
N spin function , eU correlatio n factor
Each electron has its own orbital, Multideterminant (GVB) Structure!
10
PsH – Positronium Hydride
A wave function with the correct asymptotic conditions:
Y(1,2, e ) (1 Pˆ )Y( H ) f (r )Y( Ps) g (r )
12
e
1e
Bressanini and Morosi: JCP 119, 7037 (2003)
11
Basis
In order to build compact wave functions we used
orbital functions where the cusp and the asymptotic
behavior are decoupled
1s e
ar br
1 r
2
e
ar
e
br
r 0
r
12
2-electron atoms
2
2
dr12
a
r
b
r
a
r
b
r
1 1
1 1
2 2
2 2
ˆ
exp
exp
Y (1 P12 ) exp
1 r1 1 r2 1 er12
Tails OK
2
2
r12 2
Zr
b
r
Zr
b
r
1
1 1
2
2 2
ˆ
exp
exp
Y (1 P12 ) exp
1 r1 1 r2
1 er12
Cusps OK – 3 parameters
2
r12 2
Zr
b
r
2
2 2
ˆ
exp
Y (1 P12 ) exp Zr1 exp
1 r2
1 er12
Fragments OK – 2 parameters (coalescence wave function)
13
Z dependence
Best values around for compact wave functions
D. Bressanini and G. Morosi J. Phys. B 41, 145001 (2008)
We can write a general wave function, with Z as a
parameter and fixed constants ki
Zr1 Z k 2 r12 Zr2 (k3 Z k 4 )r22 r12 / 2
exp
exp
Ψ (1,2 | Z ) (1 Pˆ12 ) exp
1 r1
1 r2
1 Z k1 r12
Tested for Z=30
Can we use this approach to larger systems?
Nodes for QMC become crucial
14
For larger atoms ?
Correlation Energy (hartree)
0
-0.1
-0.2
Exact
HF VMC
GVB VMC
HF QMC
GVB QMC
-0.3
-0.4
He
Li
Be
B
C
N
O
F
Ne
15
GVB Monte Carlo for Atoms
Correlation Energy (hartree)
0
-0.1
-0.2
Exact
HF VMC
GVB VMC
HF QMC
GVB QMC
-0.3
-0.4
He
Li
Be
B
C
N
O
F
Ne
16
Nodes does not improve
The wave function can be improved by
incorporating the known analytical structure…
with a small number of parameters
… but the nodes do not seem to improve
Was able to prove it mathematically up to N=7
(Nitrogen atom), but it seems a general feature
EVMC(YRHF) > EVMC(YGVB)
EDMC(YRHF) = EDMC(YGVB)
17
Is there anything “critical”
about the nodes of critical
wave functions?
Critical charge Zc
1
1 1
H (12 22 )
2
r1 r2 r12
2 electrons:
Critical Z for binding Zc=0.91103
Yc is square integrable
Yc
2
1
Z
<1 : infinitely many discrete bound states
1≤≤ c: only one bound state
All discrete excited state are absorbed in the
continuum exactly at =1
Their Y become more and more diffuse
19
Critical charge Zc
N electrons atom
< 1/(N-1) infinite number of discrete eigenvalues
≥ 1/(N-1) finite number of discrete eigenvalues
N-2 ≤ Zc ≤ N-1
N=3 “Lithium” atom Zc 2. As Z→ Zc YZ
2
N=4 “Beryllium” atom Zc 2.85 As Z→ Zc
Yc
2
20
Lithium atom
Spin
Spin
r13
r1 r2
r1
r12
r3
r2
YHartree Fock 0
Is r1 = r2 the exact
node of Lithium ?
Spin
r1
•
Even the exaxt node seems
to be r1 = r2, taking
different cuts (using a very
r2
r3
accurate Hylleraas expansion)
21
Varying Z: QMC versus Hylleraas
-0.7
preliminary results
The node r1=r2
seems to be
valid over a
wide range of
Energy
-0.8
-0.9
Up to c =1/2 ?
-1
Quantum Monte Carlo
Hylleraas expansion
-1.1
-1.2
0
0.1
0.2
0.3
0.4
0.5
22
Be Nodal Topology
r1+r2
r1+r2
r3-r4
r3-r4
r1-r2
YHF 0
r1-r2
YExact 0
Y 1s 2 2s 2 c 1s 2 2 p 2
23
N=4 critical charge
-0.6
N=3
Lithium
Beryllium
N=4
Energy
-0.8
-1
c
-1.2
-1.4
0
0.1
0.2
0.3
0.4
0.5
24
N=4 critical charge
-0.817
Close up
N=3
Beryllium
Lithium
N=4
Energy
-0.818
c 0.3502
-0.819
Zc 2.855
-0.82
Zc (Hogreve) 2.85
-0.821
0.348
0.349
0.35
0.351
25
N=4 critical charge node
-
0.0012
0.0008
E HF-CI
preliminary results
very close to
c=0.3502
0.0004
0
Critical Node very
close to
(r1 r2 )(r3 r4 ) 0
-0.0004
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
26
The End
Take a look at your nodes
27