Universita’ dell’Insubria, Como, Italy

Much ado about… zeroes (

of wave functions

)

Dario Bressanini http://scienze-como.uninsubria.it/bressanini Electronic Structure beyond DFT, Leiden 2004

A little advertisement

• Besides nodes, I am interested in  VMC improvement • Robust optimization • Delayed rejection VMC  Mixed 3 He/ 4 He clusters, ground and excited states  Sign problem  Other QMC topics

http://scienze-como.uninsubria.it/bressanini

2

Fixed Node Approximation

 Restrict random walk to a positive region bounded by ( approximate ) nodes.

  The energy is an upper bound Fixed Node

IS

efficient,

+

but approximation is uncontrolled  There is not ( yet ) a way to sistematically improve the nodes 

How do we build a

Y

with good nodes?

-

4

Fixed Node Approximation

circa 1950 Rediscovered by Anderson and Ceperly in the ’70s 5

Common misconception on nodes

• Nodes are not fixed by antisymmetry alone, only a 3N-3 sub-dimensional subset 6

Common misconception on nodes

• They have (almost) with Orbital Nodes.

nothing to do  It is (sometimes) orbitals possible to use nodeless 7

Common misconceptions on nodes

• A common misconception is that on a node , two like-electrons are always close . This is not true

1

Y  0

2 2

Y  0 Y  0

1

8

Common misconceptions on nodes

• Nodal theorem is NOT VALID in N-Dimensions   Higher energy states

does not

Hilbert ) mean more nodes (Courant and It is only an upper bound 9

Common misconceptions on nodes

• Not even for the same symmetry species 3 2.5

2 1.5

1 0.5

0 0 0.5

1 1.5

2 2.5

3

Courant counterexample

10

Tiling Theorem (Ceperley)

Impossible for ground state Nodal regions must have the same shape The Tiling Theorem does not say how many nodal regions we should expect 11

Nodes are relevant

• Levinson Theorem:  the number of nodes of the zero-energy scattering wave function gives the number of bound states • • Fractional quantum Hall effect Quantum Chaos Integrable system Chaotic system 12

Generalized Variational Principle

 Y

T

*

H

Y

T dR



E

0 Upper bound to ground state Higher states can be above or below

H H

2 max

H i



E N H

1

H

3 Bressanini and Reynolds, to be published 13

Nodes and Configurations

A better

Y

does not mean better nodes Why? What can we do about it?

It is necessary to get a better understanding how CSF influence the nodes.

Flad, Caffarel and Savin

14

The

(long term)

Plan of Attack

• Study the nodes of exact and good approximate trial wave functions • • Understand their properties Find a way to sistematically the nodes of trial functions improve  ...building them from scratch  …improving existing nodes 15

The Helium triplet

• First 3 S state of He is one of very few systems where we know the exact node • For S states we can write Y  Y (

r

1 ,

r

2 ,

r

12 ) • For the Pauli Principle Y (

r

1 ,

r

2 ,

r

12 )   Y (

r

2 ,

r

1 ,

r

12 ) • Which means that the node is

r

1 

r

2

or r

1 

r

2  0 16

The Helium triplet node

• • Independent of r 12 The node is more symmetric than the wave function itself • It is a polynomial in r 1 and r 2 • Present in all atoms 3

S

states of two-electron

r

1 

r

2

r

1 

r

2

r 1 r 1 r 2 r 2 r 12

 Y  0  Y  0 17

He: Other states

• Other states have similar properties • Breit ( 1930 ) showed that Y(

P

e )= (x 1

y

2 – y 1

x

2 ) f(r 1 ,r 2 ,r 12 )  2p 2 3

P

e : f( ) symmetric node = (x 1

y

2 – y 1

x

2 )  2p3p 1

P

e : f( ) antisymmetric node = (x 1

y

2 – y 1

x

2 ) (r 1 -r 2 ) • 1s2p 1

P o

: node independent from r

12

(J.B.Anderson) 18

Other He states: 1s2s 2

1

S and

2

3

S

• depend on Y  12  Y (

r

,

r

,  (or does very ) weakly) not  12 • A very good approximation of the node is •

r

1 4 

r

2 4 

const

The second triplet has similar properties

r 2 r 1

Surface contour plot of the node

r

1 5 

r

2 5 

const

22

Helium Nodes

Y

Exact



N

(

R

)

e f

(

R

) • • • Independent from r 12 Higher symmetry than the wave function Some are described by polynomials in distances and/or coordinates • The HF Y , sometimes, has the correct node, or a node with the correct (higher) symmetry • Are these general properties surfaces ?

of nodal 23

Lithium Atom Ground State

Y

RHF

 1

s

(

r

1 ) 1

s

(

r

2 ) 2

s

(

r

3 ) • The RHF node is r

1

= r 3  ( 1

s

(

r

1 ) 2

s

(

r

3 )  1

s

(

r

3 ) 2

s

(

r

1 )  1

s

(

r

2 ) if two like-spin electrons are at the same distance from the nucleus then Y =0 • • Node has higher symmetry than Y How good is the RHF node?

Y

RHF

is not very good, however its node is surprisingly good DMC( Y

RHF

) = -7.47803(5) Exact = -7.47806032

a.u.

a.u.

Lüchow & Anderson JCP 1996 Drake, Hylleraas expansion

24

•

Li

atom: Study of Exact Node

We take an “almost exact” Hylleraas expansion 250 term

r 3

• The node

r 1 = r

cuts, independent from r or r

ij 3 seems to be

, taking different 2

r 1 r 2

• a DMC simulation with

r 1

reduce the variance gives

= r 3

node and good Y to • DMC Exact -7.478061(3) -7.4780603

a.u.

a.u.

Is r 1 = r 3 the exact node of Lithium ?

25

Li

atom: Study of Exact Node

• Li exact node is more symmetric than Y Y

Hy

 

r

1

n r

2

m r

3

l r i

12

r j

13

r k

23

e



r

1  

r

2  

r

3 • At convergence, there is a delicate cancellation in order to build the node • • Crude Y has a good node (r 1 -r 3 )Exp(...) Increasing the expansion spoils by including r

ij

terms the node, 26

Nodal Symmetry Conjecture

• This observation is general: If 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 27

Beryllium Atom

Y

RHF

 1

s

(

r

1 ) 2

s

(

r

2 )  1

s

(

r

3 ) 2

s

(

r

4 )   HF predicts

4

nodal regions Bressanini et al. JCP

97

, 9200 (1992)  Node: (r 1 -r 2 )(r 3 -r 4 ) = 0  Y factors into two determinants each one “describing” a triplet Be +2 . The node is the union of the two independent nodes.

 The HF node is wrong • DMC energy 14.6576(4) • Exact energy -14.6673

Plot cuts of (r 1 -r 2 ) vs (r 3 -r 4 ) 28

Be Nodal Topology

r1+r2 r1+r2 r1-r2 Y

HF

 0 r3-r4 r3-r4 r1-r2 Y

CI

Y   1

s

2 0 2

s

2 

c

1

s

2 2

p

2 31

Be nodal topology

• Now there are only two nodal regions • It can be proved that the

exact

Be wave function has exactly two regions Node is (r 1 -r 2 )(r 3 -r 4 ) + ...

See

Bressanini, Ceperley and Reynolds

http://scienze-como.uninsubria.it/bressanini/ http://archive.ncsa.uiuc.edu/Apps/CMP/

32

Hartree-Fock Nodes

Y

HF

 1 , 2 ,...

N

 

N

  1 ,...,

N

 

N

 

J

(

r ij

) • Y HF has always,

at least

for 4 or more electrons , 4 nodal regions • • It

might

have N  ! N  ! Regions Ne atom: 5! 5! = 14400 possible regions • Li 2 molecule: 3! 3! = 36 regions How Many ?

33

Nodal Regions

Li Be B C Ne Li 2

Nodal Regions Y

HF

Y

CI

2 4 4 4 4 4 2 2 2 2 2 2 34

Nodal 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

36

Be e gas

Avoided crossings

37

r1+r2

Be model node

(

r

1 (

r

1 

r

2 )(

r

3 

r

2 )(

r

3 

r

4 ) 

r

4 )  

c c

( 

r r

12 2 13   

r

34

r

14 2   0 2

r

23  2

r

24 )  0 r3-r4 • • Second order approx.

Gives the right topology and the right shape • What's next?

r1-r2 38

Be numbers

• • • • • • • HF node GVB node Luechow & Anderson -14.667

2(2) Umrigar et al.

-14.667

18(3) Huang et al.

Casula & Sorella Exact -14.6565(2) 1s 2 2s 2 same 1s1s' 2s2s' -14.667

26(1) -14.667

28(2) -14.667

3555 +1s 2 2p 2 +1s 2 2p 2 +1s 2 2p 2 opt +1s 2 2p 2 opt • Including 1s 2 ns ms or 1s 2 np mp configurations does not improve the Fixed Node energy...

...Why?

39

Be Node: considerations

• ... ( I believe ) they give the same contribution to the node expansion • • ex: 1s 2 2s 2 and 1s 2 3s 2 have the same node ex: 2p x 2 , 2p x 3p x and 3p x 2 have the same structure

x

1

x

2

f

1 (

r

1 )

f

2 (

r

2 ) • The nodes of "useful" CSFs belong to higher and different symmetry groups than the exact Y ( 

r

1

r

12  

r

2 

r

34 )(

r

3 

r

4 )

i

ˆ 34 ( 1

s

2 2

s

2 )

i

ˆ 34 ( 1

s

2 2

p

2 )  1

s

2 2

s

2   1

s

2 2

p

2 40

-14.6668

The effect of d orbitals

1s2 2s2 = -14.6565(2) + 1s2 2p2 -14.6669

-14.6670

-14.6671

+ 1s2 3d2 -14.6672

-14.6673

-14.6674

0 Exact 0.006

0.002

Time step 0.004

41

Be numbers

• • • • • • • • HF GVB node Luechow & Anderson Umrigar et al.

Huang et al.

Casula & Sorella Bressanini et al.

Exact -14.6565(2) 1s 2 2s 2 same -14.667

2(2) 1s1s' 2s2s' +1s 2 2p 2 -14.667

18(3) -14.667

26(1) +1s +1s 2 2 2p 2p 2 2 opt -14.667

28(2) -14.667

33(7) +1s 2 2p 2 opt +1s 2 3d 2 -14.667

3555 42

CSF nodal conjecture

WARNING: Conjecture Ahead...

If the basis is sufficiently large, only built with orbitals of different configurations angular momentum and symmetry contribute to the shape of the nodes

This explains why single excitations are not useful 43

Carbon Atom: Topology

HF

1

s

2 2

s

2 2

p

2 4 Nodal Regions

GVB

 1

s

1

s

 2

s

2

s

 2

p

2

p

 (    (        4 Determinan ts 4 Nodal Regions

Adding determinants might not be sufficient to change the topology

CI

1

s

2 2

s

2 2

p

2 

c

1

s

2 2

p

4 2 Nodal Regions 44

Carbon Atom: Energy

• • • • • CSFs 1 1s 2 2s 2 2p 2 2 + 1s 2 2p 4 5 + 1s 2 2s 2p 2 3d Det.

1 2 18 Energy -37.8303(4) -37.8342(4) -37.8399(1) 83 1s 2 + 4 electrons in 2s 2p 3s 3p 3d 422 -37.8387(4) shell adding f orbitals • 7 (4f 2 + 2p 3 4f) Exact 34 -37.8407(1) -37.8450 Where is the missing energy? (g, core, optim..) 45

He

2 +

molecule

3 electrons 9-1 = 8 degrees of freedom 2  

u E

0   4 .

994598 Y

HF

 1 

g

2 1 

u

Basis: 2(1s) E=-4.9927(1) 5(1s) E=-4.9943(2) ( almost exact ) nodal surface of Y 0 r 1a , r 1b , r 2a depends on and r 2b : higher symmetry than Y 0 46

He

2 +

molecule

2 Determinants E Exact = -4.994598

E = -4.9932(2) 47

He

2 +

molecule

3 Determinants E Exact = -4.994598

E = -4.9778(3) 48

Li

2

molecule

• Adding more configuration with a small basis (double zeta STO) ...

Filippi & Umrigar JCP 1996

49

Li

2

molecule, large basis

• • Adding CFS with a larger basis ... (1  g 2 HF 2  2

g

+8

nσ g

2 GVB 8 dets -14.9919(1) -14.9914(1) -14.9907(6) 1  u 2 omitted) %CE 97.2(1) 96.7(1) 96.2(6) • + 1  2

ux

 1  2

uy

-14.9933(1) 98.3(1) +4

n

 2

ux



n

 2

uy

-14.9933(1) 98.3(1) • + 2

p z



u

2  3

p z

 2

g

-14.9952(1) Estimated n.r. limit -14.9954

99.8(1) 50

C

2

CSF 1 -75.860(1) 1 1 -75.8613(8) -75.866(2) Exact 20 36 4 32 -75.900(1) -75.9025(7) -75.8901(7) -75.900(1) -75.9255

Work in progress 1 -75.8692(5) 5(s)4(p)2(d) 12 -75.9032(8) 12 -75.9038(6) Barnett et. al.

Barnett et. al.

Filippi - Umrigar Lüchow - Fink Linear opt.

52

A tentative recipe

• Use a large Slater basis  But not too large  Try to reach HF nodes convergence • Use the right determinants...

 ...different Angular Momentum CSFs • And not the bad ones  ...types already included 53

Use a good basis

The nodes of Hartree–Fock wavefunctions and their orbitals, Chem. Phys.Lett. 392, 55 (2004) Hachmann, Galek, Yanai, Chan and, Handy 54

How to directly improve nodes?

• Fit to a functional form and optimize the parameters ( small systems ) • IF the topology is correct, use a coordinate transformation ( Linear? Feynman’s backflow ?

) Y (

R

)  Y (

T

(

R

)) 55

Conclusions

• • Nodes are worth studying !

Conjectures on nodes   have higher symmetry than Y resemble simple functions itself  the ground state has only 2 nodal volumes •  HF nodes are quite good: they “ naturally ” have these properties Recipe:  Use large basis, until HF nodes are converged  Include "different kind" of CSFs with higher angular momentum 56

Acknowledgments.. and a suggestion

Silvia Tarasco Peter Reynolds Gabriele Morosi Carlos Bunge

Take a look at

your

nodes

57

A (Nodal) 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 58