Dario Bressanini

Universita’ dell’Insubria, Como, Italy Some considerations on nodes and trial wave functions Dario Bressanini http://scienze-como.uninsubria.it/bressanini QMCI Sardagna (Trento) 2008

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

The influence on the nodes of YT • QMC currently relies on YT(R) and its nodes (indirectly) • How are the nodes YT(R) of influenced by: • The single particle basis set • The generation of the orbitals (HF, CAS, MCSCF, NO, …) • The number and type of configurations in the multidet. Expansion • The functional form of YT(R) ?

Improving YT • Current Quantum Monte Carlo research focuses on • Optimizing the energy • Adding more determinants (large number of parameters) • Exploring new trial wave function forms (moderately large number of parameters) • Pfaffians, Geminals, Backflow ... • Node are improved (but not always) only indirectly

Adding more determinants • Use a large Slater basis • Try to reach HF nodes convergence • Orbitals from MCSCF are good • 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

Li2 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 Bressanini et al. J. Chem. Phys. 123, 204109 (2005)

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: Often neglected Tails:

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. An exponential wall

GVB for atoms

Correct asymptotic structure Is there a 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 3 det large basis -14.9933(2) CI 5 det large basis -14.9952(1) E (N.R.L.) -14.9954 Improvement in the wave function but irrelevant on the nodes,

GVB in QMC • Conclusions • The quality of the wave function improves, giving better VMC energies … • … but the nodes are not changed, giving the same QMC energies • Bressanini and Morosi J. Chem. Phys. 129, 054103 (2008)

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)

What to do? • Should we be happy with the “cancellation of error”, and pursue it? • After all, the whole quantum chemistry is built on it! • If not, and pursue orthodox QMC(no pseudopotentials, no cancellation of errors, …), can we avoid thecurse of YT ?

The curse of the YT • QMC currently relies on YT(R) • Walter Kohn in its Nobel lecture (R.M.P. 71, 1253 (1999)) “discredited” 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

Numbers and insight • There is no shortage of accurate calculations for few-electron systems • −2.90372437703411959831115924519440444669690537a.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)

We need new, and different, ideas A little intermezzo (for the students)

We need new, and different, ideas • Different representations • Different dimensions • Different equations • Different potential • Radically different algorithms • Different something Research is the process of going up alleys to see if they are blind. Marston Bates

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

Use the Hypernode of Another (failed so far) example • Different dimensionality: Hypernodes • Given HY (R) = EY (R) build • The hope was that it could be better than Fixed Node

The intuitive idea was that the system could correct the wrong fixed nodes, by exploring regions where Fixed Node Fixed HyperNode Trial node Trial node Exact node Exact node Hypernodes • The energy is still an upper bound • Unfortunately, it seems to recover exactly the FN energy

Feynman on simulating nature • Nature isn’t classical, dammit, and if you want to make a simulation of Nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy” Richard Feynman 1981

Nodes • Conjectures on nodes • have higher symmetry than Y itself • resemble simple functions • the ground state has only 2 nodal volumes • HF nodes are often a god starting point Should we concentrate on nodes?

How to directly improve nodes? • Fit to a functional form and optimize the parameters (maybe for small systems) • IF the topology is correct, use a coordinate transformation

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 …. • The goal is to have only few linear parameters to optimize • Will it work???????

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

The need for the correct topology • Using Backflow alone, on a single determinant Y is not sufficient, since the topology is still wrong • More determinants are necessary (only two?)

r1+r2 r1+r2 r3-r4 r3-r4 r1-r2 r1-r2 Be Nodal Topology

Avoided crossings Be e- gas Stadium

Nodal topology • The conjecture (which I believe is true) claims that there are only two nodal volumes in the fermion ground state • See, among others: • Ceperley J.Stat.Phys63, 1237 (1991) • Bressanini and coworkers. JCP97, 9200 (1992) • Bressanini, Ceperley, Reynolds, “What do we know about wave function nodes?”, in Recent Advances in Quantum Monte Carlo Methods II, ed. S. Rothstein, World Scientfic (2001) • Mitas and coworkers PRB72, 075131 (2005) • Mitas PRL 96, 240402 (2006)

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... (any help is appreciated)

He atom with noninteracting electrons

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

Dario Bressanini