Alex Thom James Spencer EPSRC

1. Fermion Quantum Monte Carlobased on the idea of sampling “graphs”Ali AlaviUniversity of Cambridge Alex Thom James Spencer EPSRC

2. Overview Introduction and motivation Paths integrals and the Fermion sign problem FSP as a problem in “path counting” A useful combinatorial formula From path-sums to graph-sums Applications to molecular systems Towards application to periodic systems

3. i2 iP-1 i1 iP Essence of idea Express the many-electron path integral in a finite Slater Determinant basis Resum the path integral over exponentially large numbers of paths to convert path-sums => graph-sums The graphs are much more stable entities which can then be sampled. k l i j

4. A graphical, or diagrammatic, expansion of the partition function …. + + + + + 2-vertex 3-vertex Each vertex is a Slater determinant Each graph represents the sum over all paths of length P which visit all verticies of the graph Non-pertubative expansion

5. Path Integrals Consider the (thermal) density matrix: In terms of the eigenstates of the Hamiltonian: The energy can be calculated from:

6. The density matrix can be represented in real space For a single electron located at x: PE terms KE terms:harmonic spring

7. x3 x2 xP x One can simulate an electron as a ring-polymer, moving in the external field (which itself can be dynamic). Polarons [Parrinello Rahman] KE along path PE along the path

8. For N electrons - Describes closed paths which can exchange identical particle coordinates Coulomb interaction Odd permutations subtract from the Partition function: Fermion sign problem As N or b increases, there is an exponential cancellation of contributions arising from even and odd paths.

9. Slater Determinant space Let Di be a Slater determinant composed out of N orthonormal spin-orbitals [e.g. Hartree-Fock orbitals, Kohn-Sham, etc] chosen out of a set of 2M: The Di form a orthonormal set of antisymmetric functions. They are solutions to a non-interacting, or uncorrelated (mean-field) Hamiltonian H0: Full problem: Exact solutions are linear superpositions of uncorrelated determinants

10. i2 iP-1 i1 iP Paths in Slater determinant space A closed path in S.D. space A given S.D. can occur multiple times along a path

12. Hamiltonian matrix elements (Slater-Condon rules) Since H contains at most 2-body interactions: Hamiltonian connects only single and double excitations: Maximum connectivity Spin selection rule: Other symmetries may also exist Hubbard model: translational invariance; Molecules:point group symmetry Cost of calculation

13. n2 j i n1 P-2-n1-n2 j k i Search for a power series in rii Two-hop 3-hop

14. Rearranging

15. Define the nested sum: which appears in the h-hop term

16. j k i j k i l The “hop” series

17. x3 x2 x1 Using induction, one can show: AJW Thom and A Alavi, J Chem Phys, 123, 204106, (2005)

18. Residue Theorem gives

19. Some useful properties of Z-sums  Replace upper limit of sums over h to  Symmetry:

20. j=l j k i i=k l j=l j k i=k i l From “hop”-expansion to “vertex” expansion Consider the 4-hop terms: 4-vertex 3-vertex “chain” “Star” 2-vertex

21. j j k k i i Analytic summation over alternating series “Cycle function”

22. Eg. A 2-vertex graph j (For simplicity) i

23. Next compute S2:

25. G1 Gg G2 G3 Star graphs j j k k i i l l For a star-graph with g-spokes, G1,G2,…Gg attached to i

26. G12 G2 Chains graphs G3 G3 G1 G2

27. k j k i j k k j General 3-vertex graph Unfolded representation: Each spoke represents an Independent circuit on the graph Folded representation j k i Denominator is cubic polynomial in z i.e. there are 3 residues

28. Unfolded 4-vertex graph

29. Denominator is a quartic polynomial in z

30. A graphical, or diagrammatic, expansion of the partition function …. + + + + + 2-vertex 3-vertex Each vertex is a Slater determinant Each graph represents the sum over all paths of length P which visit all verticies of the graph

31. 2,3, and 4-vertex graphs + + + +

32. Monte Carlo sampling of graphs The energy can be obtained from: If graphs can be sampled with an un-normalised probability given by w(n) [G], then the energy estimator is:

33. For this to be useful, the denominator has to be well-behaved as i.e. the number of positive sampled graphs should exceed the number of negative sampled graphs in such a way that this difference is finite and does not vanish. Monitor fraction of sampled graphs which are trees, positive cyclic and negative cyclic graphs.

34. For graphs that contain the HF determinant: [Hartree-Fock energy] [Ground state energy] Approximation:Truncate series at 2-vertex, 3-vertex or higher-vertex graphs. 2 vertex: Double-excitations Number of graphs= [N2M2] 3 vertex: Quadruple excitations [N4M4] 4 vertex: Hexatuple excitations [N6M6]

35. N2 molecule

36. N2 molecule in VDZ basis Types of sampled graphs (4-vertex level)

38. N2 sampled energies (4-vertex level)

39. N2 binding curve [sampling graphs which contain the HF determinant]

41. Applications to periodic systems • Taking a plane-wave PP code (CPMD) which can solve for • KS orbitals and potential • KS virtuals • -> Use these as the basis for the vertex series • KS Hamiltonian becomes the reference (single-excitations now • contribute) • Need 2-index and 4-index integrals, which are computed on-the-fly • using FFTs (time consuming part) • Advantage: (i) Treatment of periodic systems • (ii) No BSSE • (iii) Can be used as a post-DFT method

42. Graphite (4 atom) primitive cell. 16el, BHS PP (Ec=90Ry) 2-vertex

44. Conclusions and outlook Development of QMC methods based on graphs gives a method to combat the Fermion sign problem Proof of concept for small molecular systems Major effort is now being expended on developing a periodic code….. …..perhaps to return to surface problems in due course!

45. Advantage of graph-sampling algorithm O(N2) scaling! • The observed stability at the 4-vertex level is extremely encouraging. • Current work: • Extension to higher order graphs • Improved Monte Carlo sampling • Applications to large systems

46. k m k l i j i j Ga Gb Graphs A graph a set of n distinct elements (in no particular order) with a given connectivity n distinct determinants Connectivity of graph is determined by rij

47. Compactly expressed: A set of n connected determinants Sum over all paths which visit all the determinants in G Each graph represents a sum over exponentially large numbers of paths its weight can be expected to be much better behaved than that of individual paths.

48. A graph, G, is an object on which we can represent the paths which visit all the vertices in G The weight of a given graph is the sum over all paths of length P which visit all the vertices of the graph: The prime ‘ indicates that the summation indicies must be chosen in Such a way that each vertex in G is visited at least once.

49. k m k l i j i j Ga Gb This condition ensures that the weights of two different graphs Ga and Gb (I.e. two graphs that differ in at least one vertex) do not double-count paths which visit only Ga Gb will not double-count paths which visit

50. Quantum Chemical applications Dissociation of diatomic molecules: Multiple-bond dissociation, e.g. the N2 molecule, is a major challenge to any ab initio method. Use HF orbitals generated from MOLPRO Gaussian basis set [cc-pVDZ or VTZ] Two-electron primitive integrals read in from MOLPRO output and  matrix constructed on the fly.