Universita dellInsubria Como Italy The quest for compact

Universita’ dell’Insubria, Como, Italy The quest for compact and accurate trial wave functions Dario Bressanini http: //scienze-como. uninsubria. it/bressanini Qmc in the Apuan Alps III (Vallico sotto) 2007 1

30 years of QMC in chemistry 2

The Early promises? n Solve the Schrödinger equation exactly without approximation (very strong) n Solve the Schrödinger equation with controlled approximations, and converge to the exact solution (strong) n Solve the Schrödinger equation with some approximation, and do better than other methods (weak) 3

Good for Helium studies n Thousands of theoretical and experimental papers have been published on Helium, in its various forms: Atom Small Clusters Droplets Bulk 4

Good for vibrational problems 5

For electronic structure? Sign Problem Fixed Nodal error problem 6

What to do? n Should we be happy with the “cancellation of error”, and pursue it? n If so: ´ n Is there the risk, in this case, that QMC becomes Yet Another Computational Tool, and not particularly efficient nor reliable? If not, and pursue orthodox QMC (no pseudopotentials, no cancellation of errors, …) , can we avoid the curse of YT ? 8

The curse of YT n QMC currently heavily relies on YT(R) n Walter Kohn in its Nobel lecture (R. M. P. 71, 1253 (1999)) tried to “discredit” the wave function as a non legitimate concept when N (number of electrons) is large p = parameters per variable For M=109 and p=3 N=6 M = total parameters needed The Exponential Wall 9

The curse of YT n Current research focusses on ´ ´ n optimizing the energy for moderately large expansions (good results) Exploring new trial wave function forms, with a moderately large number of parameters (good results) Is it hopeless to ask for both accurate and compact wave functions? 10

Li 2 J. Chem. Phys. 123, 204109 (2005) CSF (1 sg 2 1 su 2 omitted) E (hartree) -14. 9923(2) -14. 9914(2) -14. 9933(1) -14. 9939(2) -14. 9952(1) E (N. R. L. ) n n -14. 9954 Not all CSF are useful Only 4 csf are needed to build a statistically exact nodal surface 12

A tentative recipe n Use a large Slater basis ´ But not too large ´ Try to reach HF nodes convergence n Orbitals from CAS seem better than HF, or NO n Not worth optimizing MOs, if the basis is large enough n Only few configurations seem to improve the FN energy n Use the right determinants. . . ´ n . . . different Angular Momentum CSFs And not the bad ones ´ . . . types already included 13

Dimers Bressanini et al. J. Chem. Phys. 123, 204109 (2005) 14

Convergence to the exact Y n We must include the correct analytical structure Cusps: QMC OK 3 -body coalescence and logarithmic terms: Tails and fragments: QMC OK Usually neglected 17

Asymptotic behavior of Y n Example with 2 -e atoms is the solution of the 1 electron problem 18

Asymptotic behavior of Y n The usual form does not satisfy the asymptotic conditions A closed shell determinant has the wrong structure 19

Asymptotic behavior of Y n In general Recursively, fixing the cusps, and setting the right symmetry… Each electron has its own orbital, Multideterminant (GVB) Structure! Take 2 N coupled electrons 2 N determinants. Again an exponential wall 20

Ps. H – Positronium Hydride n A wave function with the correct asymptotic conditions: Bressanini and Morosi: JCP 119, 7037 (2003) 21

Basis n In order to build compact wave functions we used basis functions where the cusp and the asymptotic behavior are decoupled n Use one function per electron plus a simple Jastrow n Can fix the cusps of the orbitals. Very few parameters 22

GVB for atoms 23

GVB for atoms 24

GVB for atoms 25

GVB for atoms 26

GVB for atoms 27

Conventional wisdom on Y YGVB = |1 s 2 s 2 p 3| |1 s’ 2 s’| - |1 s’ 2 s 2 p 3| |1 s 2 s’| + |1 s’ 2 p 3| |1 s 2 s|- |1 s 2 s’ 2 p 3| |1 s’ 2 s| Same Node equivalent to a YUHF |f(r) g(r) 2 p 3| |1 s 2 s| EDMC(YGVB) = EDMC(Y’’RHF) 30

GVB for molecules n Correct asymptotic structure n Nodal error component in HF wave function coming from incorrect dissociation? 32

GVB for molecules Localized orbitals 33

GVB Li 2 Wave functions VMC DMC HF 1 det compact -14. 9523(2) -14. 9916(1) GVB 8 det compact -14. 9688(1) -14. 9915(1) CI 3 det compact -14. 9632(1) -14. 9931(1) GVB CI 24 det compact -14. 9782(1) -14. 9936(1) CI 5 det large basis -14. 9952(1) E (N. R. L. ) -14. 9954 Improvement in the wave function but irrelevant on the nodes, 34

Different coordinates n The usual coordinates might not be the best to describe orbitals and wave functions n In LCAO need to use large basis n For dimers, elliptical confocal coordinates are more “natural” 35

Different coordinates n Li 2 ground state n Compact MOs built using elliptic coordinates 36

Li 2 Wave functions VMC DMC HF 1 det compact -14. 9523(2) -14. 9916(1) HF 1 det elliptic -14. 9543(1) -14. 9916(1) CI 3 det compact -14. 9632(1) -14. 9931(1) CI 3 det elliptic -14. 9670(1) -14. 9937(1) E (N. R. L. ) -14. 9954 Some improvement in the wave function but negligible on the nodes, 37

Different coordinates n It might make a difference even on nodes for etheronuclei n Consider Li. H+3 the 2 ss state: n The wave function is dominated by the 2 s on Li n The node (in red) is asymmetrical n However the exact node must be symmetric HF LCAO Li H 38

Different coordinates n This is an explicit example of a phenomenon already encountered in other systems, the symmetry of the node is higher than the symmetry of the wave function n The convergence to the exact node, in LCAO, is very slow. n Using elliptical coordinates is the right way to proceed n Future work will explore if this effect might be important in the construction of many body nodes HF LCAO Li H 39

Playing directly with nodes? n It would be useful to be able to optimize only those parameters that alter the nodal structure n A first “exploration” using a simple test system: He 2+ n The nodes seem to be smooth and “simple” n Can we “expand” the nodes on a basis? 40

He 2+: “expanding” the node n It is a one parameter Y !! Exact 41

“expanding” nodes n This was only a kind of “proof of concept” n It remains to be seen if it can be applied to larger systems ´ Writing “simple” (algebraic? ) trial nodes is not difficult …. ´ Waierstrass theorem ´ The goal is to have only few linear parameters to optimize ´ Will it work? ? ? ? 42

Conclusions n The wave function can be improved by incorporating the known analytical structure… with a small number of parameters n … but the nodes do not seem to improve n It seems more promising to directly “manipulate” the nodes. 44

A QMC song. . . He deals the cards to find the answers the sacred geometry of chance the hidden law of a probable outcome the numbers lead a dance Sting: Shape of my heart 45

Just an example n Try a different representation n Is some QMC in the momentum representation ´ Possible ? And if so, is it: ´ Practical ? ´ Useful/Advantageus ? ´ Eventually better than plain vanilla QMC ? ´ Better suited for some problems/systems ? ´ Less plagued by the usual problems ? 46

The other half of Quantum mechanics The Schrodinger equation in the momentum representation Some QMC (GFMC) should be possible, given the iterative form Or write the imaginary time propagator in momentum space 47

Better? n For coulomb systems: n There are NO cusps in momentum space. Y convergence should be faster n Hydrogenic orbitals are simple rational functions 48

Avoided nodal crossing n At a nodal crossing, Y and Y are zero n Avoided nodal crossing is the rule, not the exception n Not (yet) a proof. . . In the generic case there is no solution to these equations If has 4 nodes has 2 nodes, with a proper 54

He atom with noninteracting electrons 55

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How to directly improve nodes? n “expand” the nodes and optimize the parameters n IF the topology is correct, use a coordinate transformation 59

Coordinate transformation n Take a wave function with the correct nodal topology n Change the nodes with a coordinate transformation (Linear? Backflow ? ) preserving the topology Miller-Good transformations 60