Mathematical representations of quantum states

Transcription

1 Mathematical representations of quantum states Eric CANCES Ecole des Ponts and INRIA, Paris, France IPAM, Los Angeles, Septembre 13-16, 2016

2 Outline of the tutorial 2 1. Quantum states of many-particle systems 2. First-principle molecular simulation (mathematical viewpoint) 3. Representations of electronic states

3 1 - Quantum states of many-particle systems pure states vs mixed states time-dependent states vs steady (or stationary) states separable states vs entangled states bound states vs scattering states ground states vs excited states

4 1 - Quantum states of many-particle systems 4 First principles of quantum mechanics An isolated quantum system is described by a state space H (a complex Hilbert space); a Hamiltonian H (a self-adjoint operator on H); other observables (s.a. op. on H) allowing one to connect theory and exper. The system is said to be in a pure state at time t if its state can be completely characterized by a wavefunction Ψ(t) H such that Ψ(t) H = 1. Time-dependent Schrödinger equation i dψ (t) = HΨ(t) dt Time-dependent Schrödinger equation The steady states are of the form Ψ(t) = f(t)ψ, f(t) C, ψ H Hψ = Eψ, E R, ψ H = 1, f(t) = e iet/

5 1 - Quantum states of many-particle systems 5 Quantum mechanics for one-particle systems Consider a particle of mass m subjected to an external potential V ext : state space: H = L 2 (R 3, C) (spin is omitted for simplicity); Hamiltonian: H = 2 2m + V ext (self-adjoint operator on H). Ψ(t, r) 2 : probability density of observing the particle at point r at time t ˆ R 3 Ψ(t, r) 2 dr = Ψ(t) 2 H = 1. Time-dependent Schrödinger equation i dψ dt 2 (t) = HΨ(t) i Ψ(t, r) = t 2m Ψ(t, r) + V ext(r)ψ(t, r)

6 1 - Quantum states of many-particle systems 6 Time-independent Schrödinger equation Hψ = Eψ ψ = 1 2 2m ψ(r) + V ext(r)ψ(r) = Eψ(r) ˆ R 3 ψ(r) 2 dr = 1 Typical spectrum of the Hamiltonian H = 2 2m + V ext for 1 e systems Ground state Scattering states (continuous spectrum) Excited states e2 Ex.: V ext (r) = (Hydrogen atom), σ(h) = 4πε 0 r { E Ryd n 2 } n N [0, + [.

7 1 - Quantum states of many-particle systems 7 Physical meaning of the discrete energy levels Example of the hydrogen atom 2 2m e Ψ(x) E n = E Ryd n, n 2 N, E Ryd = m e 2 e2 Ψ(x) = EΨ(x) 4πε 0 x ( ) e 2 2, λ m n = 8π c 4πε 0 E Ryd ( 1 n 1 ) 1 2 m 2 Balmer series (nm): λ exp 6 2 = , λexp 5 2 = , λexp 4 2 = , λexp 3 2 = λ 6 2 = , λ 5 2 = , λ 4 2 = , λ 3 2 =

8 1 - Quantum states of many-particle systems 8 Physical meaning of the discrete energy levels Example of the hydrogen atom 2 2m e Ψ(x) E n = E Ryd n, n 2 N, E Ryd = m e 2 e2 Ψ(x) = EΨ(x) 4πε 0 x ( ) e 2 2, λ m n = 8π c 4πε 0 E Ryd ( 1 n 1 ) 1 2 m 2 Balmer series (nm): λ exp 6 2 = , λexp 5 2 = , λexp 4 2 = , λexp 3 2 = = λ corr. 6 2 = , λ corr. 5 2 = , λ corr. 4 2 = , λ corr.

9 1 - Quantum states of many-particle systems 9 On the physical meaning of point and continuous spectra Theorem (RAGE, Ruelle 69, Amrein and Georgescu 73, Enss 78). Let H be a locally compact self-adjoint operator on L 2 (R d ). [Ex.: the Hamiltonian of the hydrogen atom satisfies these assumptions.] Let H p = Span {eigenvectors of H} and H c = H p. [Ex.: for the Hamiltonian of the hydrogen atom, dim(h p ) = dim(h c ) =.] Let χ BR be the characteristic function of the ball B R = { r R d r < R }. Then (φ 0 H p ) ε > 0, R > 0, t 0, (φ 0 H c ) R > 0, lim T + 1 T ˆ T 0 (1 χ BR )e ith/ 2 φ 0 ε; L 2 χ BR e ith/ 2 φ 0 dt = 0. L2

10 1 - Quantum states of many-particle systems 9 On the physical meaning of point and continuous spectra Theorem (RAGE, Ruelle 69, Amrein and Georgescu 73, Enss 78). Let H be a locally compact self-adjoint operator on L 2 (R d ). [Ex.: the Hamiltonian of the hydrogen atom satisfies these assumptions.] Let H p = Span {eigenvectors of H} and H c = H p. [Ex.: for the Hamiltonian of the hydrogen atom, dim(h p ) = dim(h c ) =.] Let χ BR be the characteristic function of the ball B R = { r R d r < R }. Then (φ 0 H p ) ε > 0, R > 0, t 0, (φ 0 H c ) R > 0, lim T + 1 T ˆ T 0 (1 χ BR )e ith/ 2 φ 0 ε; L 2 χ BR e ith/ 2 φ 0 dt = 0. L2 H p set of bound states, H c set of scattering states.

11 1 - Quantum states of many-particle systems 10 Important remarks 1. Quantum mechanics is an intrinsically probabilistic theory: according to the Copenhagen interpretation, the fact that two different experiments made on two identical systems in the same pure state do not necessarily lead to the same result is not due to an incomplete knowledge of the state. Historical milestones: Einstein-Podolski-Rosen paradox ( 35); Bell inequalities ( 64); Aspect s experiments ( 81). 2. The set of pure states of an isolated quantum system with state space H is diffeomorphic to the projective space P (H) (H \ {0})/C. This is the reason why the two wavefunctions ψ and e iα ψ, with ψ H, ψ = 1 and α R, represent exactly the same state.

12 Entangled states exhibit correlations that have no classical analogue. For this reason, they can be used to certify the principles of quantum mechanics, and also play key roles in quantum information and quantum computing Quantum states of many-particle systems 11 Composite quantum systems If a system can be decomposed into two distinguishable quantum subsystems A and B, then the state space of this system is H AB = H A H B, where H A and H B are the state spaces of A and B respectively. Separable (or factored) and entangled states Consider a composite quantum system with state space H AB = H A H B. Definition. A pure state ψ AB H AB is called separable if there exist ψ A H A and ψ B H B such that ψ AB = ψ A ψ B. Otherwise, it is called entangled. The quantity of entanglement of a state can be measured by entropies of reduced density matrices.

13 1 - Quantum states of many-particle systems 12 Composite quantum systems If a system can be decomposed into two indistinguishable quantum subsystems A and A, then the state space of this system is H AA = H A H A (symmetrized tensor product) if A is a boson, H A H A (antisymmetrized tensor product) if A is a fermion, where H A is the state space of A. According to the spin-statistics theorem, bosons are quantum objects with integer spins, fermions are quantum objects with half-integer spins. The definition and quantification of entanglement for composite systems made of identical bosons or identical fermions is still a field of research.

14 1 - Quantum states of many-particle systems 13 Quantum mechanics for two-particle systems State space: H L 2 (R 3, C) L 2 (R 3, C) L 2 (R 6, C) Ψ(t, r 1, r 2 ) 2 : probability density of observing at time t the particle 1 at r 1 and the particle 2 at r 2 Symmetry constraints two different particles: H = L 2 (R 3, C) L 2 (R 3, C) two identical bosons (e.g. two C 12 nuclei): H = L 2 (R 3, C) L 2 (R 3, C) Ψ(t, r 2, r 1 ) = Ψ(t, r 1, r 2 ) two identical fermions (e.g. two electrons): H = L 2 (R 3, C) L 2 (R 3, C) Ψ(t, r 2, r 1 ) = Ψ(t, r 1, r 2 ) (Pauli principle) ˆ ˆ ˆ density ρ(t, r) = Ψ(t, r, r 2 ) 2 dr 2 + Ψ(t, r 1, r) 2 dr 1 = 2 Ψ(t, r, r 2 ) 2 dr 2 R 3 R 3 R 3

15 1 - Quantum states of many-particle systems 14 Quantum mechanics for N-particle systems Consider N particles of masses m 1,, m N subjected to an external potential V ext (r) and pair-interaction potentials W ij (r i, r j ). State space: H L 2 (R 3, C) L 2 (R 3, C) L 2 (R 3N, C) Ψ(t, r 1,, r N ) 2 : probability density of observing at time t the particle 1 at r 1, the particle 2 at r 2,... Time-independent Schrödinger equation N i=1 2 2m i ri + N V ext (r i ) + i=1 1 i<j N W ij (r i, r j ) Ψ(r 1,, r N ) = E Ψ(r 1,, r N ) 3N-dimensional linear ellipic eigenvalue problem

16 1 - Quantum states of many-particle systems 15 Ground state of N non-interacting identical particles of mass m subjected to an external potential V ext (r) N 2 N N H = 2m r i + V ext (r i ) = h ri i=1 i=1 i=1 h φ i = ε i φ i, ε 1 ε 2 ε N ˆ R 3 φ i φ j = δ ij 0 ε F h = 2 2m + V ext Bosonic ground state: ψ(r 1,, r N ) = N=5 N φ 1 (r i ), ρ(r) = N φ 1 (r) 2 Fermionic gr. st.: ψ(r 1,, r N ) = 1 N! det(φ i (r j )), ρ(r) = i=1 N φ i (r) 2 i=1

17 1 - Quantum states of many-particle systems 16 Γ(t) = The pure-state formalism presented so far is not always sufficient: when a N-particle system is confined in a cavity and may exchange energy with its environment, it is in general not in a pure state, but in a mixed state represented by a density matrix + i=1 n i (t) ψ i (t) ψ i (t), ψ i (t) ψ j (t) = δ ij, 0 n i (t) 1, + i=1 n i (t) = 1. At thermal equilibrium at temperature T, the state of the system is Γ = Z 1 exp ( βh) where Z = Tr (exp ( βh)) and β = 1 k B T. The set K N of N-body density matrices is a weakly- closed bounded convex subset B(H N ) and the pure states are the extreme points of K; when the system may exchange particles with its environment, or when particles can be created or destroyed, N-particle state spaces must be replaced by Fock spaces; when the system contains infinitely many-particles (e.g. disordered crystals), states can be conveniently described in terms of suitable linear forms on the C -algebra of admissible observables.

18 Molecular biology Nanotechnology 2 - First-principle molecular simulation (mathematical viewpoint) Chemistry Materials science

19 2 - First-principle molecular simulation 18 Key observation A molecule is a set of M nuclei and N electrons. The state space H L 2 (R 3(M+N), C) and the Hamiltonian of the molecule can be deduced from its chemical formula: M 1 N 1 N M H = Rk 2m k 2 z k r i r i R k + 1 r i r j + z k z l R k R l k=1 i=1 i=1 k=1 1 i<j N Atomic units: = 1, m e = 1, e = 1, 4πε 0 = 1. 1 k<l M This Hamiltonian is free of empirical parameters specific to the system. The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be solved. (Dirac, 1929)

20 2 - First-principle molecular simulation 19 Ionization energy of Helium (Korobov & Yelkhovsky 01): He + hν He + + e Ground state energy of He E=h ν E (e c ) = h ν exp. : MHz ( 97) MHz ( 98) Ground state energy of He +

21 2 - First-principle molecular simulation 19 Ionization energy of Helium (Korobov & Yelkhovsky 01): He + hν He + + e Ground state energy of He E=h ν E (e c ) = h ν exp. : MHz ( 97) MHz ( 98) Ground state energy of He +

22 2 - First-principle molecular simulation 19 Ionization energy of Helium (Korobov & Yelkhovsky 01): He + hν He + + e Ground state energy of He E=h ν E (e c ) = h ν exp. : MHz ( 97) MHz ( 98) calc.: MHz Ground state energy of He + H He = 1 2m R 1 2 r r 2 H He + = 1 2m R 1 2 r 1 2 r 1 R 2 r 2 R + 1 r 1 r 2 2, m = a.u. r 1 R

23 2 - First-principle molecular simulation 19 Ionization energy of Helium (Korobov & Yelkhovsky 01): He + hν He + + e Ground state energy of He E=h ν E (e c ) = h ν Ground state energy of He + exp. : MHz ( 97) MHz ( 98) calc.: MHz MHz (R.C.) H He = 1 2m R 1 2 r r 2 H He + = 1 2m R 1 2 r 1 2 r 1 R 2 r 1 R 2 r 2 R + 1 r 1 r 2 + Breit terms + Breit terms

24 2 - First-principle molecular simulation 20 Example: computation of some properties of the water molecule (H 2 O) A water molecule consists of M = 3 atomic nuclei (1 oxygen + 2 hydrogens) and N = 10 electrons in Coulomb interaction. Such a system can be described by the laws of quantum mechanics (many-body Schrödinger equation) and statistical physics. The only parameters of these models are a few fundamental constants of physics (atomic units) = 1, m e = 1, e = 1, ε 0 = (4π) 1, c , k B = the charges and masses of the hydrogen and oxygen (16) nuclei z H = 1, z O = 8, m H = , m16 O = Born-Oppenheimer strategy (based on the fact that m e /m nuc 1): Step 1: definition of the potential energy surfaces (elec. struct. calc.) Step 2: analysis of the potential energy surfaces.

25 2 - First-principle molecular simulation 21 Electronic problem for a given nuclear configuration {R k } 1 k M Ex: water molecule H 2 O M = 3, N = 10, z 1 = 8, z 2 = 1, z 3 = 1 V ne {R k } (r) = M k=1 z k r R k 1 2 N ri + i=1 N i=1 V ne {R k } (r i) + 1 i<j N 1 Ψ(r 1,, r N ) = E Ψ(r 1,, r N ) r i r j Ψ(r 1,, r N ) 2 probability density of observing electron 1 at r 1, electron 2 at r 2,... Warning: in this lecture, spin is omitted for simplicity

26 2 - First-principle molecular simulation 21 Electronic problem for a given nuclear configuration {R k } 1 k M Ex: water molecule H 2 O M = 3, N = 10, z 1 = 8, z 2 = 1, z 3 = 1 V ne {R k } (r) = M k=1 z k r R k 1 2 N ri + i=1 N i=1 V ne {R k } (r i) + 1 i<j N 1 Ψ(r 1,, r N ) = E Ψ(r 1,, r N ) r i r j Ψ(r 1,, r N ) 2 probability density of observing electron 1 at r 1, electron 2 at r 2,... p S N, Ψ(r p(1),, r p(n) ) = ε(p)ψ(r 1,, r N ), (Pauli principle)

27 2 - First-principle molecular simulation 21 Electronic problem for a given nuclear configuration {R k } 1 k M Ex: water molecule H 2 O M = 3, N = 10, z 1 = 8, z 2 = 1, z 3 = 1 V ne {R k } (r) = M k=1 z k r R k 1 2 N ri + i=1 N i=1 V ne {R k } (r i) + 1 i<j N 1 Ψ(r 1,, r N ) = E Ψ(r 1,, r N ) r i r j Ψ(r 1,, r N ) 2 probability density of observing electron 1 at r 1, electron 2 at r 2,... p S N, Ψ(r p(1),, r p(n) ) = ε(p)ψ(r 1,, r N ), (Pauli principle) ˆ Ψ L 2 = 1, ρ Ψ (r) = N Ψ(r, r 2,, r N ) 2 dr 2 dr N R 3(N 1)

28 2 - First-principle molecular simulation 22 Spectrum of the electronic hamiltonian N H {R k} 1 N N = 2 r i V{R ne k } (r i)+ i=1 i=1 Zhislin s theorem: if N σ(h {R k} N ) = 1 i<j N 1 r i r j on H N := N L 2 (R 3, C) (Pauli principle) M z k (neutral or positively charged system), then k=1 { E {R k} 0 E {R k} 1 E {R k} 2 } [Σ {R k}, + ). Ground state Excited states Ε {R k } 0 {R } Σ k Essential spectrum

29 2 - First-principle molecular simulation 23 Step 1: definition of the potential energy surfaces W n (R 1,, R M ) = E {R k} n + 1 k<l M z k z l R k R l σ (H ) N {R k} W 0

30 2 - First-principle molecular simulation 24 Step 2: analysis of the potential energy surfaces Born-Oppenheimer approximation adiabatic approximation m e /m nuc 1 semiclassical approximation on the nuclear dynamics: 0 W 0 σ (H ) N {R k}

31 2 - First-principle molecular simulation 24 Step 2: analysis of the potential energy surfaces Born-Oppenheimer approximation adiabatic approximation m e /m nuc 1 semiclassical approximation on the nuclear dynamics: 0 W 0 σ (H ) N {R k} First-principle molecular dynamics m k d 2 R k dt 2 (t) = R k W 0 (R 1 (t),, R M (t)), 1 k M The nuclei behave as point-like classical particle interacting via the effective M-body potential W 0.

32 2 - First-principle molecular simulation 24 Step 2: analysis of the potential energy surfaces Born-Oppenheimer approximation adiabatic approximation m e /m nuc 1 semiclassical approximation on the nuclear dynamics: 0 W 0 σ (H ) N {R k} Global minima of W 0 : equilibrium configurations of the system H H O pm

33 2 - First-principle molecular simulation 24 Step 2: analysis of the potential energy surfaces Born-Oppenheimer approximation σ (H ) N adiabatic approximation m e /m nuc 1 {R k} semiclassical approximation on the nuclear dynamics: 0 W 0 Vibration frequencies (harm. approx.) R k (t) = R 0 k + y k (t) m k d 2 y k,i dt 2 = M l=1 3 j=1 2 W 0 R k,i R l,j (R 0 )y l,j

34 2 - First-principle molecular simulation 24 Step 2: analysis of the potential energy surfaces Born-Oppenheimer approximation σ (H ) N adiabatic approximation m e /m nuc 1 {R k} semiclassical approximation on the nuclear dynamics: 0 W 0 Vibration frequencies (harm. approx.) R k (t) = R 0 k + y k (t) m k d 2 y k,i dt 2 = M l=1 3 j=1 2 W 0 R k,i R l,j (R 0 )y l,j infrared spectrum

35 2 - First-principle molecular simulation 24 Step 2: analysis of the potential energy surfaces Born-Oppenheimer approximation adiabatic approximation m e /m nuc 1 semiclassical approximation on the nuclear dynamics: 0 W 0 σ (H ) N {R k} Vertical transition energies: visible spectrum (color) ultraviolet spectrum X spectrum ionization energy

36 2 - First-principle molecular simulation 24 Step 2: analysis of the potential energy surfaces Born-Oppenheimer approximation σ (H ) N adiabatic approximation m e /m nuc 1 {R k} semiclassical approximation on the nuclear dynamics: 0 W 0 E a Local minima: (meta)stable states (reactants and products) Critical points of W 0 with Morse index 1: transition states OH + H H 2 O k TST = Π3N 6 i=1 νi Re Π 3N 7 i=1 ν TS,+ i e E a/k B T (large deviation theory).

37 2 - First-principle molecular simulation 25 Main practical issue: compute the ground state electronic energy and density for a given nuclear configuration. Hartree Fock N body electronic Schrodinger equation Wavefunction methods Density functional theory (DFT) Quantum Monte Carlo Single reference methods: MPn, CI, CC,... Multi reference methods: MCSCF, MRCC,... Thomas Fermi (orbital free) : TF, TFW,... Kohn Sham : Hartree, X α, LDA, GGA,... Variational MC Diffusion MC For brevity, we restrict ourselves to Hartree-Fock and Kohn-Sham (mean-field, one-particle effective models); single reference 2nd order methods: MP2, CISC, CCSD.

38 3 - Representation of electronic states

39 3 - Representation of electronic states 27 Reduced density matrix and p-body densities The wavefunction Ψ (resp. the density matrix Γ) contains all the information about the state. This is way too much for practical applications! p-body reduced density matrix (p-rdm) associated with Ψ H N γ p,ψ (r 1,, r p ; r 1,, r ( ) ˆ p) N = Ψ(r p 1,, r p, r p+1,, r N )Ψ(r 1,, r p, r p+1,, r N ) dr p+1 dr N R 3(N p) γ p,ψ should be seen as the integral kernel of a bounded self-adjoint linear operator γ p,ψ : H p H p, called the p-rdo associated Ψ. p-body density associated with a wavefunction Ψ H N ρ p,ψ (r 1,, r p ) = γ p,ψ (r 1,, r p ; r 1,, r p ) p-rdo associated with mixed states Γ = + i=1 n i Ψ i Ψ i γ p,γ = + i=1 n i γ p,ψi (linear map!)

40 3 - Representation of electronic states 28 Key properties of reduced density matrices and p-body densities Expectation values of non-local (resp. local) p-body observables can be computed from the p-rdm (resp. the p-body density). In particular, ( Ψ H {R k} N Ψ = Tr 1 ) ˆ ˆ 2 γ 1,ψ + ρ 1,Ψ V ne R 3 {R k } +1 ρ 2,Ψ(r 1, r 2 ) dr 1 dr 2. 2 R 3 R 3 r 1 r 2 If Ψ is the Slater determinant of orthonormal orbitals (φ 1,, φ N ), then N γ 1,Ψ (r, r ) = φ i (r)φ i (r ), and all the γ p,ψ, ρ p,ψ are simple functions of γ 1,Ψ. In particular, i=1 ρ 1,Ψ (r) = γ 1,Ψ (r, r) and ρ 2,Ψ (r 1, r 2 ) = ρ 1,Ψ (r 1 )ρ 1,Ψ (r 2 ) γ 1,Ψ (r 1, r 2 ) 2. Unfortunately, no practical characterization of the sets {(γ 1, ρ 2 ) Ψ H N s.t. γ 1 = γ 1,Ψ, ρ 2 = ρ 2,Ψ } {(γ 1, ρ 2 ) Γ K N s.t. γ 1 = γ 1,Γ, ρ 2 = ρ 2,Γ } is available for numerical simulation (N-representability problem).

41 3 - Representation of electronic states 29 Discretization of the (spinless) one-electron state space One-electron wavefunctions are usually discretized either in atomic orbitals (quantum chemistry: H 1 = L 2 (R 3, C)) or in planewaves (solid state physics and materials science: H 1 = L 2 per(γ, C)). Other discretization methods include grid, finite-element, wavelets, and discontinuous Galerkin methods. In both atomic orbitals (AO) and planewave (PW) methods, one-electron wavefunctions are expanded in a basis χ = (χ 1,, χ Nb ) (H 1 ) N b as where, typically, φ(r) = N b µ=1 α µ χ µ (r) N b 2N 10N for AO basis sets (depending on the nuclear coord.); N b 100N 1000N for PW basis sets, N being the number of electrons in the system (in the simulation cell for PW).

42 3 - Representation of electronic states 30 Representation of discretized Hartree-Fock and Kohn-Sham states Let χ = (χ 1,, χ Nb ) be the chosen discretization basis of H 1 and H χ 1 := Span(χ 1,, χ Nb ). Discretized HF and KS states are Slater det. build with orbitals from H χ 1 ( ˆ ) N b (φ 1,, φ N ) (H χ 1 )N, φ i φ j = δ ij φ i (r) = C µi χ µ (r), C S χ C = I N where S χ is the overlap matrix of the basis χ = (χ 1,, χ Nb ): ˆ Sµν χ = χ µχ ν Important remark: for any unitary matrix U U(N), C and CU give rise to the same wavefunction Ψ (properties of Slater determinants). They therefore represent the same state. This is related to the fact that γ 1,Ψ (r, r ) = N b µ,ν=1 µ=1 D µν χ µ (r)χ ν (r) with D = CC.

43 3 - Representation of electronic states 31 Orbital vs density matrix formulation of HF and KS problems E {R k},hf/ks 0,χ = inf = inf { { E {R k},hf/ks } χ (CC ), C C Nb N, C S χ C = I N } (D), D C N b N b, D = D = DS χ D, Tr(S χ D) = N E {R k},hf/ks χ E {R k},hf χ (D) = Tr(h {Rk},χ D) Tr(Gχ (D)D) linear quadratic E {R k},ks χ (D) = Tr(h {Rk},χ D) Tr(J χ (D)D) + Eχ xc (D) Core Hamiltonian matrix: h {R k},χ µν = 1 2 linear quadratic nonlinear ˆ R 3 χ µ χ ν M ˆ z k k=1 χ µ (r) χ ν (r) R 3 r R k dr

44 3 - Representation of electronic states 32 The set of discretized HF and KS states in the basis χ = (χ 1,, χ Nb ) is therefore diffeomorphic to where S χ N and Gχ N Remarks: S χ N /U(N) Gχ N respectively denote the Stiefel and Grassmann manifolds S χ N = { C C N b N C S χ C = I N }, G χ N = { D C N b N b D = D = DS χ D, Tr (S χ D) = N }. the manifolds S χ N /U(N) and Gχ N have the same dimension N(N b N); from a geometrical point of view, the Grassmann manifold G χ N is the submanifold of L(C N b) consisting of the rank-n orthogonal projectors when C N b is endowed with the scalar product defined by S χ (and therefore with the canonical inner product when the basis χ is orthonormal); when both the basis set functions χ j and the orbitals are real-valued, the result holds with C replaced with R and U(N) replaced by O(N), the group of real orthogonal matrices.

45 3 - Representation of electronic states 33 Exponential map The manifold G χ N is a nonlinear object. It can however be parametrized in the following way: a reference density matrix D 0 G χ N being given, the "exponential map" f :C N (Nb N) G χ N ( A f(a) = (S χ ) 1 0N A 2 exp A 0 Nb N ) ( (S χ ) 2D 1 0 (S χ ) 1 0N A 2 exp A 0 Nb N ) (S χ ) 1 2 defines a local chart of G χ N centered at f(0) = D 0. Note that f is surjective but not injective.

46 3 - Representation of electronic states 34 Discretized of N-body wavefunctions and p-rdos If Ψ H χ N, then γ p,ψ χ Hp : Hp χ Hp χ and Ψ(r 1,, r N ) = γ p,ψ = 1 µ 1 < < µ p N b 1 ν 1 < < ν p N b 1 κ 1 < <κ N N b c κ1,,κ N ( χκ1 χ κn ) (r1,, r N ) D (p) µ 1,,µ p ;ν 1,,ν p χ µ1 χ µp χ ν1 χ νp. Discretized 1-body density matrix: if Ψ H χ N, then with γ 1,Ψ (r, r ) = N b µ,ν=1 D µ,ν χ µ (r)χ ν (r ), D C N b N b, D = D, 0 DS χ D D, Tr(S χ D) = N. and DS χ D = D if and only if Ψ is a Slater determinant.

47 3 - Representation of electronic states 35 Second quantization formalism Fock space F := + N=0 H N, H 0 = C, H 1 = L 2 (R 3, C), H N = Annihilation and creation operators N H1. φ H 1, a(φ) : H N H N 1, a (φ) : H N H N+1, a (φ) = (a(φ)), Ψ N H N, (a(φ)ψ N )(r 1,, r N 1 ) = N ˆ R 3 φ(r) Ψ N (r, r 1,, r N 1 ) dr. a A(H 1, B(F)), a B(H 1, B(F)), Note that if φ H χ 1 and Ψ N H χ N, then a(φ)ψ N H χ N 1 and a (φ)ψ N H χ N+1.

48 3 - Representation of electronic states 36 Discretized N-electron state spaces and Fock space Let χ = (χ 1,, χ Nb ) be the chosen discretization basis of H 1 and H χ 1 := Span(χ 1,, χ Nb ). The discrete N-electron state space is then defined for all 1 N N b as N H χ N := ( ) H χ 1 and dim(h χ N ) = Nb curse of dimensionality! N A (non-orthogonal in general) basis of H χ N is ψ µ1,µ 2,,µ N (r 1,, r N ) = 1 N! det(χ µi (r j )), 1 µ 1 < µ 2 < < µ N N b. The corresponding discretization of the Fock space F := F χ := N b N=0 + N=0 H χ N, with Hχ 0 = C, so that dim(f χ ) = 2 N b. H N then is

49 3 - Representation of electronic states 37 Excitations with respect to a reference Slater determinant A Hartree-Fock (or Kohn-Sham) calculation in the basis set χ provides a set of N b orthonormal molecular orbitals belonging to H χ 1, among which N occupied orbitals (φ i ) 1 i N, (N b N) unoccupied (or virtual) orbitals (φ a ) N+1 a Nb. The Slater determinant constructed with the N occupied orbitals Φ 0 (r 1,, r N ) = 1 det(φ i (r j )) H χ N N! is the Hartree-Fock (or Kohn-Sham) ground state in the basis set χ. Denoting by a ij = a(φ ij ) and a a k = a (φ ak ) the annihilation and creation operators associated with the occupied and virtual orbitals φ ij and φ ak respectively, an orthonormal basis of H χ N is obtained by collecting: the reference Slater determinant Φ 0 ; the single-excitation Slater determinants Φ a 1 i 1 = a a 1 a i1 Φ 0 ; the double-excitation Slater determinants Φ a 1a 2 i 1 i 2 = a a 1 a a 2 a i1 a i2 Φ 0 ;...

50 3 - Representation of electronic states 38 Excitation operators single-excitation: T a 1 i 1 = a a 1 a i1, 1 i 1 N, N + 1 a 1 N b di-excitation: T a 1a 2 i 1 i 2 = a a 1 a a 2 a i1 a i2, 1 i 1 < i 2 N, N + 1 a 1 < a 2 N b... Excitation subspaces single-double (SD) excitation subspaces V 2 = Span { I, T a 1 i 1, T a 1a 2 } i 1 i Φ0 2 W 2 = Span { T a 1 i 1, T a 1a 2 } i 1 i 2 single-double (SDT) excitation subspaces H χ N B(H χ N )... V 3 = Span { I, T a 1 i 1, T a 1a 2 i 1 i 2, T a 1a 2 a 3 } i 1 i 2 i Ψ0 3 H χ N W 3 = Span { T a 1 i 1, T a 1a 2 i 1 i 2, T a 1a 2 a 3 } i 1 i 2 i 3 B(H χ N )

51 3 - Representation of electronic states 39 Single-double configuration interaction (CISD): simple Galerkin approx. Find (E, Ψ) R V 2 such that Ψ V 2, Ψ H N Ψ = E Ψ Ψ Ψ = 1 Linear eigenvalue problem of size dim(v 2 ) 1 4 N 2 (N b N) 2 Single-double coupled-cluster (CCSD): based on the property { e T Φ 0, T W N } = {Ψ HN Φ 0 Ψ = 1}. { Find (E, T ) R W2 such that Ψ V 2, Ψ H N e T Φ 0 = E Ψ e T Φ 0 { Find (E, T ) R W2 such that Ψ V 2, Ψ e T H N e T Φ 0 = E Ψ Φ 0 Quartic algebraic equation with dim(w 2 ) 1 4 N 2 (N b N) 2 real unknowns

52 Take-home messages and conclusion

53 Take-home messages and conclusion 41 Relevant representation levels of quantum states The wavefunction Ψ (resp. the density matrix Γ) contains all the information about the state. This is way too much for practical applications! Expectation values of non-local (resp. local) p-body observables can be computed from the p-rdm (resp. the p-body density). Representation of HF and KS states HF and KS states are Slater determinants. After discretization in a basis set χ = (χ 1,, χ N b ), the set of HS or KS states is diffeomorphic to S χ N /U(N) Gχ N where S χ N and Gχ N respectively denote the Stiefel and Grassmann manifolds S χ N = { C C N b N C S χ C = I N }, G χ N = { D C N b N b D = D = DS χ D, Tr (S χ D) = N }. Note that for atomic basis sets, the overlap matrix S χ µν = χ µχ ν depends on the atomic positions. The exponential map provides local charts of the manifold G χ N.

54 Take-home messages and conclusion 42 Representation of CI and CC states in the second quantization formalism, CI and CC states are represented by order-n tensors with n = 4 for CISD and CCSD, n = 6 for CISDT and CCSDT, n = 8 for CISDTQ, CCSDTQ, etc. Ex. : for CCSD t a i and tab ij, 1 i < j N, N + 1 a < b N b low-rank tensor methods may allow one to store and manipulate these tensors at a much lower cost in terms of memory and computational time.