Title: Trial wave function construction and the nodes of trial and exact wave functions in
1Trial wave function construction and the nodes of
trial and exact wave functions in Quantum Monte
Carlo
Dario Bressanini Universita dellInsubria,
Como, Italy http//www.unico.it/dario
CECAM 2003 Fermion Sign Problem - Lyon
2Nodes and the Sign Problem
- Fixed-node QMC is efficient. If only we could
have the exact nodes - or at least a systematic way to improve the
nodes ... - we could bypass the sign problem
- How do we build a Y with good nodes?
3Nodes
- What do we know about wave function nodes?
- Very little ....
- NOT fixed by (anti)symmetry alone.Only a 3N-3
subset
- Very very few analytic examples
- Nodal theorem is NOT VALID
- Higher energy states does not mean more nodes
(Courant and Hilbert ) - They have (almost) nothing to do with Orbital
Nodes. It is possible to use nodeless orbitals.
4Tiling Theorem (Ceperley)
Impossible for ground state
The Tiling Theorem does not say how many nodal
regions we should expect
5Fixed Node Approximation
circa 1950
6Nodes and Configurations
It is necessary to get a better understanding how
CSF influence the nodes. Flad, Caffarel and
Savin
7The (long term) Plan of Attack
- Study the nodes of exact and good approximate
trial wave functions - Understand their properties
- Find a way to sistematically improve the nodes of
trial functions - Find a way to parametrize the nodes using simple
functions, and optimize the nodes directly
minimizing the Fixed-Node energy
8The Helium Triplet
- First 3S state of He is one of very few systems
where we know exact node - For S states we can write
- Which means that the node is
9The Helium Triplet
- Independent of r12
- The node is more symmetric than the wave function
itself - It is a polynomial in r1 and r2
- Present in all 3S states of two-electron atoms
10He 3S a look at non-physical regions
- Consider Y(r1,r2,q12) defined in all space
- A node in a non-physical regions appears. Using a
simple trial function...
11He 3S a look at non-physical regions
- Using accurate trial functions...
- Expanding Y at second order in (0,0)
- Y (10-6 0.001 (r1r2))(r1-r2)...
12He 3S a look at non-physical regions
- If we turn off the e-e interaction we observe the
same feature (r1r2)(r1-r2)/2...
- There is no apparent reason why even the exact
wave function should be - Y c (r1r2)(r1-r2)...
- It seems the nodal structure of the exact wave
function resembles the independent electron case
13Other He states 1s2s 2 1S and 2 3S
14He Other states
- 1s2s 3S (r1-r2) f(r1,r2,r12)
- 1s2p 1P o node independent from r12
(J.B.Anderson) - 2p2 3P e Y (x1 y2 y1 x2) f(r1,r2,r12)
- 2p3p 1P e Y (x1 y2 y1 x2) (r1-r2)
f(r1,r2,r12) - 1s2s 1S node independent from r12
- 1s3s 3S node independent from r12
15Helium Nodes
- Independent from r12
- More symmetric than the wave function
- Some are described by polynomials in distances
and/or coordinates - The same node is present in different states
- The HF Y, sometimes, has the correct node, or a
node with the correct (higher) symmetry - Are these general properties of nodal surfaces ?
16Lithium Atom Ground State
- The RHF node is r1 r3
- if two like-spin electrons are at the same
distance from the nucleus then Y 0 - This is the same node present in the He 3S
- Again, node has higher symmetry than Y
- How good is the RHF node?
- YRHF is not very good, however its node is
surprisingly good (might it be the exact one?) - DMC(YRHF ) -7.47803(5) a.u. Lüchow Anderson
JCP 1996 - Exact -7.47806032 a.u. Drake, Hylleraas
expansion
17Li atom Study of Exact Node
- We take an almost exact Hylleraas expansion 250
term
- The node seems to ber1 r3, taking different
cuts, independent from r2 or rij
- a DMC simulation with r1 r3 node and good Y to
reduce the variance gives - DMC -7.478061(3) a.u. Exact -7.4780603
a.u.
Is r1 r3 the exact node of Lithium ?
18Li atom Study of Exact Node
- Li exact node is more symmetric than Y
- At convergence, there is a delicate cancellation
in order to build the node - Crude Y has a good node (r1-r3)Exp(...)
- Increasing the expansion spoils the node, by
including rij terms
19Nodal Symmetry Conjecture
- This observation is generalIf the symmetry of
the nodes is higher than the symmetry of Y,
adding terms in Y might decrease the quality of
the nodes (which is what we often see).
WARNING Conjecture Ahead...
Symmetry of nodes of Y is higher than symmetry of
Y
20Beryllium Atom
- HF predicts 4 nodal regions Bressanini et al.
JCP 97, 9200 (1992) - Node (r1-r2)(r3-r4) 0
- Y factors into two determinants each one
describing a triplet Be2. The node is the
union of the two independent nodes.
- The HF node is wrong
- DMC energy -14.6576(4)
- Exact energy -14.6673
21Be beyond Restricted Hartree-Fock
- Hartree-Fock Y is not the most general single
particle approximation - Try a GVB wave function (4 determinants)
VMC energy improves, s2(H) improves... but still
the same node (r1-r2)(r3-r4) 0
22Be CI expansion
- What happens to the HF node in a good CI
expansion?
- In 9-D space, the direct product structure opens
up
Node is (r1-r2)(r3-r4) ...
23Be Nodal Topology
24Be nodal topology
- Now there are only two nodal regions
- It can be proved that the exact Be wave function
has exactly two regions
See Bressanini, Ceperley and Reynolds http//www.
unico.it/dario/ http//archive.ncsa.uiuc.edu/App
s/CMP/
Node is (r1-r2) (r3-r4) ???
25Be model node
- Second order approx.
- Gives the right topology and the right shape
- What's next?
26Be numbers
- HF node -14.6565(2) 1s2 2s2
- GVB node same 1s1s' 2s2s'
- Luechow Anderson -14.6672(2) 1s2 2p2
- Umrigar et al. -14.66718(3) 1s2 2p2
- Huang et al. -14.66726(1) 1s2 2p2 opt
- Casula Sorella -14.66728(2) 1s2 2p2 opt
- Exact -14.6673555
- Including 1s2 ns ms or 1s2 np mp configurations
does not improve the Fixed Node energy... - ...Why?
27Be Node considerations
- ... (I believe) they give the same contribution
to the node expansion - ex 1s22s2 and 1s23s2 have the same node
- ex 2px2, 2px3px and 3px2 have the same structure
- The nodes of "useful" CSFs belong to higher and
different symmetry groups than the exact Y
28The effect of d orbitals
29Be numbers
- HF -14.6565(2) 1s2 2s2
- GVB node same 1s1s' 2s2s'
- Luechow Anderson -14.6672(2) 1s2 2p2
- Umrigar et al. -14.66718(3) 1s2 2p2
- Huang et al. -14.66726(1) 1s2 2p2 opt
- Casula Sorella -14.66728(2) 1s2 2p2 opt
- Bressanini et al. -14.66733(7) 1s2 3d2
- Exact -14.6673555
30CSF nodal conjecture
WARNING Conjecture Ahead...
If the basis is sufficiently large, only
configurations built with orbitals of different
angular momentum and symmetry contribute to the
shape of the nodes
This explains why single excitations are not
useful
31Carbon Atom
Is it possible to change the topology of the
nodal hypersurfaces by adding particular CSFs or
do they merely generate deformations?
Both!
32Carbon Atom
- CSFs Det. Energy
- 1 1s22s2 2p2 1 -37.8303(4)
- 2 1s2 2p4 2 -37.8342(4)
- 5 1s2 2s 2p23d 18 -37.8399(1)
- 83 1s2 4 electrons in 2s 2p 3s 3p 3d
shell) 422 -37.8387(4) - adding f orbitals
- 7 (4f2 2p34f) 34 -37.8407(1)
- Exact -37.8450
- Where is the missing energy? (g, core, optim..)
33Li2 molecule
- Adding more configuration with a small basis
(double zeta STO)...
34Li2 molecule, larger basis
- Adding CFS with a larger basis ... (1sg2 1su2
omitted)
- GVB 8 dets -14.9907(6) 96.2(6)
Estimated n.r. limit -14.9954
35O2
- Small basis
- 1 Det. -150.268(1) Umrigar et al.
- 7 Det. -150.277(1) .....................
- Exact -150.3268
- Large basis
- 1 Det. -150.2850(6) Tarasco, work in progress
- 2 Det. -150.2873(7) ..............................
....
36Hartree-Fock Nodes
- YHF has always, at least, 4 nodal regions for 4
or more electrons - It might have Na! Nb! Regions
- Ne atom 5! 5! 14400 possible regions
- Li2 molecule 3! 3! 36 regions
How Many ?
37Nodal Regions
38Nodal Topology Conjecture
WARNING Conjecture Ahead...
The HF ground state of Atomic and Molecular
systems has 4 Nodal Regions, while the Exact
ground state has only 2
39Conclusions
- Exact or good nodes (at least for simple systems)
seem to - depend on few variables
- have higher symmetry than Y itself
- resemble simple functions
- Possible explanation on why HF nodes are quite
good they naturally have these properties - Use large basis, until HF nodes are converged
- Include "different" CSFs
- Has the ground state only 2 nodal volumes?
40Acknowledgments
- Silvia Tarasco
- Peter Reynolds
- Gabriele Morosi
- Carlos Bunge
41A (QMC) song...
He deals the cards to find the answers the secret
geometry of chance the hidden law of a probable
outcome the numbers lead a dance
Sting Shape of my heart