Tensor product approximation and thenumerical solution of the Electronic

Schrodinger equation

R. Schneider (TUB Matheon)

John von Neumann Lecture – TU Munich, 2012

Introduction

Electronic Schrodinger Equation

Quantum mechanics

Goal: Calculation of physical and chemicalproperties on a microscopic (atomic)

length scale.

B E.g. atoms, molecules, clusters, solidsB E.g. chemical behaviour, bonding

energies, ionization energies,conduction properties, essentialmaterial properties

Electronic structure calculationReduction of the problem to the computation of an electronicwave function Ψ for given fixed nuclei.

The electronic Schrodinger equation, describes the stationarynonrelativistic behaviour of system of N electrons in anelectrical field

HΨ = EΨ.

Electronic structure determines e.g.

B bonding energies,reactivity

B ionization energiesB conductivity,B in a wider sense

molecular geometry,-dynamics,...

of atoms, molecules, solidsetc.

Electronic Schrodinger equationN nonrelativistic electrons +

Born Oppenheimer approxi-

mationHΨ = EΨ

The Hamilton operator in atomic units

H = −12

∑i

∆i −N∑i

K∑ν=1

Zν|xi − aν |

+12

N∑i 6=j

1|xi − xj |

acts on anti-symmetric wave functions (Pauli principle)

Ψ(x1, s1, . . . , xN , sN) ∈ R , xi = (xi , si) ∈ R3 × {±12},

Ψ(. . . ; xi , si ; . . . ; xj , sj ; . . .) = −Ψ(. . . ; xj , sj ; . . . ; xi , si ; . . .)

Variational formulationInput: position aν of the ν’s atom, nuclear charge Zν , number ofelectrons N.

Output: We are mainly interested in the ground-state energy,i.e. the lowest eigenvalue in the configuration spaceΨ ∈ V = H1(R3 × {±1

2})N ∩

∧Ni=1 L2(R3 × {±1

2}) .E0 = min〈Ψ,Ψ〉=1〈HΨ,Ψ〉 , Ψ = argmin〈Ψ,Ψ〉=1〈HΨ,Ψ〉

I Energy surfaces E = E(a1, . . .ak ). (Add nuclear repulsionpotential

∑ν 6=µ

ZνZµ2|aν−aµ| !)

I atomic forces ∂E∂aν ⇒ molecular geometry

I bonding and ionizationenergies etc.

I these quantities are (small) differences E0,a − E0,b

I accuracy is limited due to neglecting relativistic andnon-Born-Openheimer effects

.. is beyond the present presentation

Example in quantum chemistry

O2 binding to hemoglobin modeled by a Fe-prophyrin complex(heme) 1Eh ≈ 27, 2114eV Hartree

| | | | |6.3e-2 mEh6.3e-16.3Rea tion barrierBinding energy

6.3e46.3e5O2�heme�

Kopie.jpg

Facts to know

The ES has been welll studied in Analysis or MathematicalPhysics

I Kato, ... (.. 60..), The energy space is : V : H1((R3×{±12})

N′)i.e. the Hamilton operator maps H : V → V ′ boundedly.

I HVZ-Theorem ( ..60..), E0 is an eigenvalue of finitemultiplicity: −∞ < E0 < infσess(H) if N ′ ≤ Z :=

∑Kν=1 Zν

I Agmon ( .. 70 .. ), exponential decay at infinity:Ψ(x) = O(e−a|x|) if |x| → ∞.

I Kato, T.- von Ostenhoff & T. Soerensen ... (98),cusp-singularities:e.g. electron-nucleon (e-N) cusp O(|xi − aν |)and electron-electron (e-e) cusp O(|xi − xj |)

I Yserentant (03) mixed regularity Ψ ∈ H1,s, s ≤ 12 , resp. 1

Basic Problem - Curse of dimensionsI linear eigenvalue problem, but extremely high-dimensionalI + anti-symmetry constraints + lack of regularity.I traditional approximation methods (FEM, Fourier series, polynomials, MRA etc.):

approximation error in R1: . n−s , s- regularity , R3N′ : . n−s3N′ , (s < 5

2 ) with nDOFs

I Curse of dimensionalityI in principle, deterministic approximation methods are scaling exponentially with

N.I Nondeterministic methods: Quantum Monte Carlo methods have problems with

fermions

For large systems N′ >> 1 ( N′ > 1) the electronic Schrodinger equation seems to be

intractable! But 70 years of impressive progress has been awarded by the Nobel price

1998 in Chemistry: Kohn, Pople.

For extended systems, the method of choice is DensityFunctional Theory (DFT). However this will be defered to mycolloquium lecture.In the sequel, we will consider only discretized equations andhigh dimensional functions.For sake of simplicity, we keep all vector spaces finitedimensional.

Further motivations: PDE’s in higher dimensionsEquations describing complex systems with multi-variatesolution spaces, e.g.

B stationary/instationary Schrodinger type equations

i~∂

∂tΨ(t , x) = (−1

2∆ + V )︸ ︷︷ ︸H

Ψ(t , x), HΨ(x) = EΨ(x)

describing quantum-mechanical many particle systemsB stochastic DEs (SDEs) and the Fokker-Planck equation,

∂p(t , x)∂t

=d∑

i=1

∂

∂xi

(fi(t , x)p(t , x)

)+

12

d∑i,j=1

∂2

∂xi∂xj

(Bi,j(t , x)p(t , x)

)describing mechanical systems in stochastic environment,

B chemical master equations, parametric PDEs, machinelearning, . . .

Solutions depend on x = (x1, . . . , xd ), where usually, d >> 3!

Further motivations: PDE’s in higher dimensionsEquations describing complex systems with multi-variatesolution spaces, e.g.

B stationary/instationary Schrodinger type equations

i~∂

∂tΨ(t , x) = (−1

2∆ + V )︸ ︷︷ ︸H

Ψ(t , x), HΨ(x) = EΨ(x)

describing quantum-mechanical many particle systemsB stochastic DEs (SDEs) and the Fokker-Planck equation,

∂p(t , x)∂t

=d∑

i=1

∂

∂xi

(fi(t , x)p(t , x)

)+

12

d∑i,j=1

∂2

∂xi∂xj

(Bi,j(t , x)p(t , x)

)describing mechanical systems in stochastic environment,

B chemical master equations, parametric PDEs, machinelearning, . . .

Solutions depend on x = (x1, . . . , xd ), where usually, d >> 3!

Further motivations: PDE’s in higher dimensionsEquations describing complex systems with multi-variatesolution spaces, e.g.

B stationary/instationary Schrodinger type equations

i~∂

∂tΨ(t , x) = (−1

2∆ + V )︸ ︷︷ ︸H

Ψ(t , x), HΨ(x) = EΨ(x)

describing quantum-mechanical many particle systemsB stochastic DEs (SDEs) and the Fokker-Planck equation,

∂p(t , x)∂t

=d∑

i=1

∂

∂xi

(fi(t , x)p(t , x)

)+

12

d∑i,j=1

∂2

∂xi∂xj

(Bi,j(t , x)p(t , x)

)describing mechanical systems in stochastic environment,

B chemical master equations, parametric PDEs, machinelearning, . . .

Solutions depend on x = (x1, . . . , xd ), where usually, d >> 3!

Further motivations: PDE’s in higher dimensionsEquations describing complex systems with multi-variatesolution spaces, e.g.

B stationary/instationary Schrodinger type equations

i~∂

∂tΨ(t , x) = (−1

2∆ + V )︸ ︷︷ ︸H

Ψ(t , x), HΨ(x) = EΨ(x)

describing quantum-mechanical many particle systemsB stochastic DEs (SDEs) and the Fokker-Planck equation,

∂p(t , x)∂t

=d∑

i=1

∂

∂xi

(fi(t , x)p(t , x)

)+

12

d∑i,j=1

∂2

∂xi∂xj

(Bi,j(t , x)p(t , x)

)describing mechanical systems in stochastic environment,

B chemical master equations, parametric PDEs, machinelearning, . . .

Solutions depend on x = (x1, . . . , xd ), where usually, d >> 3!

Further motivations: PDE’s in higher dimensionsEquations describing complex systems with multi-variatesolution spaces, e.g.

B stationary/instationary Schrodinger type equations

i~∂

∂tΨ(t , x) = (−1

2∆ + V )︸ ︷︷ ︸H

Ψ(t , x), HΨ(x) = EΨ(x)

describing quantum-mechanical many particle systemsB stochastic DEs (SDEs) and the Fokker-Planck equation,

∂p(t , x)∂t

=d∑

i=1

∂

∂xi

(fi(t , x)p(t , x)

)+

12

d∑i,j=1

∂2

∂xi∂xj

(Bi,j(t , x)p(t , x)

)describing mechanical systems in stochastic environment,

B chemical master equations, parametric PDEs, machinelearning, . . .

Solutions depend on x = (x1, . . . , xd ), where usually, d >> 3!

Setting - Tensors of order dGoal: Generic perspective on methods for high-dimensionalproblems, i.e. problems posed on tensor spaces,

V :=⊗d

i=1 Vi , today: V =⊗d

i=1 Rn = R(nd )

Notation: (x1, . . . , xd ) 7→ U = U(x1, . . . , xd ) ∈ V

Main problem:

dim V = O(nd ) – Curse of dimensionality!

e.g. n = 100,d = 10 10010 basis functions, coefficient vectors of 800× 1018 Bytes = 800 Exabytes

Approach: Some higher order tensors can be constructed(data-) sparsely from lower order quantities.

As for matrices, incomplete SVD:

A(x1, x2) ≈r∑

k=1

σk(uk (x1)⊗ vk (x2)

)

Setting - Tensors of order dGoal: Generic perspective on methods for high-dimensionalproblems, i.e. problems posed on tensor spaces,

V :=⊗d

i=1 Vi , today: V =⊗d

i=1 Rn = R(nd )

Notation: (x1, . . . , xd ) 7→ U = U(x1, . . . , xd ) ∈ V

Main problem:

dim V = O(nd ) – Curse of dimensionality!

e.g. n = 100,d = 10 10010 basis functions, coefficient vectors of 800× 1018 Bytes = 800 Exabytes

Approach: Some higher order tensors can be constructed(data-) sparsely from lower order quantities.

As for matrices, incomplete SVD:

A(x1, x2) ≈r∑

k=1

σk(uk (x1)⊗ vk (x2)

)

Setting - Tensors of order dGoal: Generic perspective on methods for high-dimensionalproblems, i.e. problems posed on tensor spaces,

V :=⊗d

i=1 Vi , today: V =⊗d

i=1 Rn = R(nd )

Notation: (x1, . . . , xd ) 7→ U = U(x1, . . . , xd ) ∈ V

Main problem:

dim V = O(nd ) – Curse of dimensionality!

e.g. n = 100,d = 10 10010 basis functions, coefficient vectors of 800× 1018 Bytes = 800 Exabytes

Approach: Some higher order tensors can be constructed(data-) sparsely from lower order quantities.

As for matrices, incomplete SVD:

A(x1, x2) ≈r∑

k=1

σk(uk (x1)⊗ vk (x2)

)

Setting - Tensors of order dGoal: Generic perspective on methods for high-dimensionalproblems, i.e. problems posed on tensor spaces,

V :=⊗d

i=1 Vi , today: V =⊗d

i=1 Rn = R(nd )

Notation: (x1, . . . , xd ) 7→ U = U(x1, . . . , xd ) ∈ V

Main problem:

dim V = O(nd ) – Curse of dimensionality!

e.g. n = 100,d = 10 10010 basis functions, coefficient vectors of 800× 1018 Bytes = 800 Exabytes

Approach: Some higher order tensors can be constructed(data-) sparsely from lower order quantities.

Canonical decomposition for order-d-tensors:

U(x1, . . . , xd ) ≈r∑

k=1

σk(⊗d

i=1 ui,k (xi)).

Further application of TP approximationI approximation of multi-parametric functionsI e.g. machine learningI signal processingI parametric PDE’s, (boundary value problems with

uncertain coefficients)I high dimensional integrationI computational financeI quantum information theoryI algebraic geometry (see book Landsberg)I vector tensorizationI ...

Although these issues could be not be discussed here in detail.Major challenge: How to avoid the curse of dimensionality?I Tensor product approximation seems to be promising?I Question: what is an appropriate generalization of SVD to

higher order tensors?

Content and time lineI Part I - Tensor Approximation

1. Classical and novel tensor formats2. Tensor networks, TT and HT tensors - - Recovery,

approximation and optimization3. ALS and MALS for TT and related formats – Matrix product

states and DMRG algorithm4. Geometry of the hierarchical Tucker format (including TT

and Tucker)5. Dynamical low rank approximation6. vector tensorization and QTT tensors

I Part 2 - Numerical solution of the electronic Schrodingerequation

1. Slater determinants,Full CI and discrete Fock spaces,2. Hartree Fock approximations and multi-configurational SCF3. Matrix product states (TT tensors) and DMRG4. Coupled Cluster method I5. Coupled Cluster method II and simplifications: MP2, RPA

and CEPA6. Reduced density matrices and Greens functions

Thank youfor your attention.

References:T. Helgaker, P. Jorgensen, J. Olsen Molecular Electronic-Structure Theory, Wiley; 1 edition (2000)

P.G. Ciarlet, J.-L. Lions, C. Le Bris, Handbook of Numerical Analysis: Special Volume: Computational ChemistryNorth-Holland, 2003

A. Szabo, N. S. Ostlund Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, DoverBooks on Chemistry, Dover Publications; New edition edition (July 2, 1996) (too old, but perfect introduction)

J. M. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, vol. 128, AMST. G. Kolda, B. W. Bader, Tensor decompositions and applications, SIAM Review Vol. 51, 3, 455-500,

W. Hackbusch, Tensor spaces and numerical tensor calculus,SSCM, vol. 42, to appear, Springer, 2012.

Ch. Lubich, From quantum to classical molecular dynamics: reduced models and numerical analysis, EuropeanMath. Soc., 2008.

U. Schollwoeck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005)O. Legeza, T. Rohwedder and R. Schneider, Numerical approaches for high-dimensional PDE’s for quantumchemistry , in Encyclopedia of Applied and Computational Mathematics, B. Engquist edt., Springer, to appear

H. Yserentant, Regularity and Approximability of Electronic Wave Functions, Lecture Notes in Mathematics, 2000,Springer, 2010.