Dario Bressanini

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

30 years of QMC in chemistry

The Early promises? • Solve the Schrödinger equation exactly withoutapproximation(very strong) • Solve the Schrödinger equation with controlled approximations, and converge to the exact solution (strong) • Solve the Schrödinger equation with some approximation, and do better than other methods (weak)

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

Good for vibrational problems

For electronic structure? Sign Problem Fixed Nodal error problem

What to do? • Should we be happy with the “cancellation of error”, and pursue it? • If so: • 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 thecurse of YT ?

The curse of YT • QMC currently heavily relies on YT(R) • 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 For M=109 andp=3 N=6 p = parameters per variable M = total parameters needed The Exponential Wall

The curse of YT • Current research focusses on • 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?

Li2 J. Chem. Phys. 123, 204109 (2005) CSF E (hartree) (1sg2 1su2 omitted) -14.9923(2) -14.9914(2) -14.9933(2) -14.9933(1) -14.9939(2) -14.9952(1) E (N.R.L.) -14.9954 • Not all CSF are useful • Only 4 csf are needed to build a statistically exact nodal surface

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

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

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

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

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

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

PsH – Positronium Hydride • A wave function with the correct asymptotic conditions: Bressanini and Morosi: JCP 119, 7037 (2003)

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

GVB for atoms

Same Node Conventional wisdom on Y YGVB = |1s 2s 2p3| |1s’ 2s’| - |1s’ 2s 2p3| |1s 2s’| + |1s’ 2s’ 2p3| |1s 2s|- |1s 2s’ 2p3| |1s’ 2s| Node equivalent to a YUHF |f(r) g(r) 2p3| |1s 2s| EDMC(YGVB) = EDMC(Y’’RHF)

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

GVB for molecules Localized orbitals

GVB Li2 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,

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

Different coordinates • Li2 ground state • Compact MOs built using elliptic coordinates

Li2 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,

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

HF LCAO H Li Different coordinates • 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 • The convergence to the exact node, in LCAO, is very slow. • Using elliptical coordinates is the right way to proceed • Future work will explore if this effect might be important in the construction of many body nodes

Playing directly with nodes? • It would be useful to be able to optimize only those parameters that alter the nodal structure • A first “exploration” using a simple test system: • The nodes seem to be smooth and “simple” • Can we “expand” the nodes on a basis? He2+

He2+: “expanding” the node • It is a one parameter Y !! Exact

“expanding” nodes • This was only a kind of “proof of concept” • 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???????

Conclusions • 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  • It seems more promising to directly “manipulate” the nodes.

A QMC song... He deals the cards to find the answers the sacredgeometry of chance thehidden lawof a probable outcome the numbers lead a dance Sting: Shape of my heart

Just an example • Try a different representation • 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 ?

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

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

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

He atom with noninteracting electrons

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

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

Dario Bressanini