Time-dependent linear-response variational Monte Carlo. Bastien Mussard bastien.mussard@colorado.edu https://mussard.github.io/ Julien Toulouse julien.toulouse@upmc.fr Sorbonne University, Paris (web) Cyrus Umrigar cyrusumrigar@cornell.edu Cornell University (web) Sandeep Sharma sandeep.sharma@colorado.edu University of Colorado at Boulder (web)
Motivation [2 / 11] The excitation energies can be calculated by linear response of the ground-state wave function to a time-dependent perturbation (TDDFT ; TDHF and other costly wavefunction methods ; CIS is the simplest way to approach it). QMC methods were originally formulated for ground state problems and their extension to excited states is not straightforward. State-average VMC [Filippi,Zaccheddu,Buda JCTC 5 (2009)] State-specific VMC [Zimmerman,Toulouse,Zhang,Musgrave,Umrigar JCP (2009)] This work is an LR-VMC extension of VMC to the calculation of electronic excitation energies and oscillator strengths using time-dependent linear-response theory. We exploit the formal analogy existing between the linear method for wave-function optimization and the generalised eigenvalue equation of linear-response theory. Very similar derivations and analogy were obtained by E. Neuscamman, in a EOM-VMC context. [Zhao,Neuscamman JCTC 12 (2016)]
The linear method for wavefunction optimization [1/2] [3 / 11] The wavefunction is parametrized with p = {α, κ, c, ξ} (Jastrow parameters α, orbital rotation κ, CI c and basis set ξ coefficients) : Ψ(p) = Ĵ(α)e ˆκ(κ) c I C I Consider the first-order expansion around the current parameters p 0 ( p = p p 0 ) of the intermediate-normalized wavefunction Ψ(p) = Ψ(p) Ψ(p) : Ψ lin (p) = + p i Ψ i where : = Ψ(p 0 ) Ψ i = Ψ(p) p i p=p 0 The optimal parameters p are found by minimizing the energy E lin corresponding to the wavefunction Ψ lin (p) : E lin = min p and iteratively updating the parameters p 0 + p p 0. Ψ lin (p) Ĥ Ψ lin (p) Ψlin (p) Ψlin (p) (1)
The linear method for wavefunction optimization [2/2] [4 / 11] Application of this variational principle amounts to solving the generalized eigenvalue equation estimated as : ( ) ( ) ( ) ( ) E0 g (1) R T 1 1 0 T 1 = E H p lin 0 S p g L where : - g L,i = Ψ i Ĥ and g R,i = Ĥ Ψ i are the left- and right-gradients. They are identical except on a finite MC sample. - H ij = Ψ i Ĥ Ψ j is the Hamiltonian matrix in the basis of the first-order derivatives. This estimator is non-symmetric. - S ij = Ψ i Ψj is the overlap matrix in this basis. - E 0 is the energy of the current and E lin is the energy of Ψ lin. - The sizes of the whole matrices are (1 + N param ).(1 + N param ) - [Toulouse,Umrigar JCP 126 (2007)] The linear-method can be seen as a stabilized Newton method. In quantum chemistry, also known as Super-CI or Augmented Hessian method.
Linear response equations [1/2] [5 / 11] The excited-state energies are calculated from the optimized ground state by linear-response time-dependent perturbation theory : Ĥ(t) = Ĥ + γ ˆV (t) Consider the expansion around the current parameters p 0 ( p(t) = p(t) p 0 ) of the intermediate-normalized wavefunction Ψ(p(t)) = Ψ(p(t)) Ψ(p(t)) : Ψ(p(t)) = + p i (t) Ψ i + 1 2 pi (t) p j (t) Ψ ij + where : = Ψ(p 0 ) Ψ i = Ψ(p(t)) p i (t) p(t)=p 0 Ψ ij = 2 Ψ(p(t)) p i (t) p j (t) p(t)=p 0 The approximate ground state evolves in time via the parameters p(t) according to the Dirac-Frenkel variational principle : Ψ(p(t)) Ĥ(t) i t min Ψ(p(t)) (2) p(t) Ψ(p(t)) Ψ(p(t))
Linear response equations [2/2] [6 / 11] Application of this variational principle (in first order in γ (linear response), keeping only first-order terms in p(t), and in the limit of a vanishing perturbation) yields (2) A p(t) + B p(t) = is p(t) t where : - Aij = Ψ i Ĥ E 0 Ψ j = H ij E 0 S ij - B ij = Ψ ij Ĥ contains (without any surprise) the second-order derivatives of the wavefunction with respect to its parameters. Looking for free-oscillation solutions of ω excitation (de-excitation) energies : p(t) = Xe iωt + Y e iωt we arrive at the linear-response equation in the form of a non-hermitian generalized eigenvalue equation : ( ) ( ) ( ) ( ) A B X S 0 X (2) B A = ω Y 0 S. Y - The sizes of the whole matrices are (2N param ).(2N param ) - The solutions come in pairs : (ω, (X, Y)) and ( ω, (Y, X ))
Linear response equations : TDA approximation [7 / 11] The TDA neglects B, i.e. : AX = ωsx Note : at the ground-state minimum, the energy gradient is zero, the generalised eigenvalue equation of the linear method is equivalent to the TDA equation : ( E0 0 T 0 H ) ( ) 1 p ( ) ( ) 1 0 T 1 = E lin 0 S p H p = E lin S p [remember : A = H E 0 S and : ω = E lin E 0 ] AX = ωsx Some remarks ( ) A B - If the response is calculated on a stable ground state the matrix is positive-definite : the excitation energies are real (as they should be). - In practice, the approximate ground state can be unstable (spin-symmetry breaking), or the response can be calculated in a bigger parameter space the minimization was done in unexploitable imaginary excitation energies. B A [7 / 11]
Realization in VMC [8 / 11] E 0 = E L Ψi Ψi g L,i = Ψ i Ĥ = Ψ i Ĥ Ψ i Ĥ = E L g R,i = Ĥ Ψ i = Ĥ Ψ i Ĥ Ψ i = S ij = Ψ i Ψj H ij = Ψ i Ĥ Ψ j =... ĤΨi Ψi Ψ j = Ψ i Ψ j Ψ i Ψ j = Ψi Ψi E L E L Ψj A ij = H ij E 0 S ij =... B ij = Ψ ij Ĥ =... E L (R) = Ĥ(R)(R) (R) E L,i (R) Ψ i (R) (R) Ψ ij (R) (R)
Beryllium : Excitation energies [9 / 11] HF orbitals, VB1[5s, 1p] VB2[6s, 2p, 1d], Jastrow (en,ee,een, 24 parameters) Error on excitation energy (Hartree) 0.3 0.2 0.1 0.0 CIS LR-VMC/TDA(j) LR-VMC/TDA(j+o) Error on excitation energy (Hartree) 0.04 0.02 0.00-0.02 CIS LR-VMC/TDA(o) VB1 VB2 VB1 VB2-0.04 VB1 VB2 VB1 VB2 (a) 2s3s( 1 S) (b) 2s4s( 1 S) (c) 2s2p( 1 P) (d) 2s2p( 3 P) (a) Jastrow adds significant physics Further effect of orbitals Big effect of the basis with CIS (b) CIS fails dramatically Jastrow reduces the error by 2 Ha in VB1 Poor description of the 4s orbital (c) Lowest energy excitation Good results for CIS LR better than CIS for VB1 (d) CIS greatly underestimates LR are very close Little basis effect in this case
Beryllium : Oscillator strength of the singlet 2s2p ( 1 P) [10 / 11] 0.00 Error on oscillator strength -0.20-0.40-0.60-0.80-1.00 VB1 CIS LR-VMC/TDA(o) VB2 - LR-VMC/TDA(o) oscillator strengths seem more sensitive to the basis set compared to the CIS oscillator strengths. - The inclusion of the Jastrow factor does not improve the oscillator strength.
Perspectives [11 / 11] LR-VMC generally outperforms CIS for excitation energies and is thus a promising approach for calculating electronic excited-state properties. - In this LR-VMC approach, after optimizing only one ground-state wave function, one can easily calculate several excitation energies of different spatial or spin symmetry (v. state-specific or state-average excited-state QMC methods). - The presence of the Jastrow factor in LR-VMC allows one to explicitly treat dynamical correlation (v. similar linear-response quantum chemistry methods). - A disadvantage is that the excitation energies are much more sensitive than the ground state energy to the quality of the optimized ground state wave function (also in other linear-response quantum chemistry methods, but : stochastic). Coming up : - systematic study on a set of molecules - larger basis sets - multideterminantal wavefunction - using the full response, beyond TDA (effect of the B matrix) - derivatives with respect to the exponents of the Slater functions. - ideas to lower some statistical variation