Exponential Parameterization of Wave Functions for Quantum Dynamics

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Citation: J. Chem. Phys. 149, 134110 (2018); doi: 10.1063/1.5049344 View online: https://doi.org/10.1063/1.5049344 View Table of Contents: http://aip.scitation.org/toc/jcp/149/13 Published by the American Institute of Physics

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Exponential parameterization of wave functions for quantum dynamics: Time-dependent Hartree in second quantization Niels Kristian Madsen, Mads Bøttger Hansen, Alberto Zoccante,a) Kasper Monrad, Mikkel Bo Hansen, and Ove Christiansenb) Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark (Received 20 July 2018; accepted 17 September 2018; published online 4 October 2018)

We derive equations for describing the time evolution of variational wave functions in linear and exponential parameterization with a second-quantization (SQ) formulation. The SQ formalism covers time-dependent Hartree (TDH), while exact states and approximate vibrational configuration inter- action wave functions are described using state-transfer operators. We present detailed expressions for efficient evaluation of TDH in linear (L-TDH) and exponential (X-TDH) parametrization and an efficient implementation supporting linear scaling with respect to the number of degrees of freedom M when the Hamiltonian operator contains a constant number of terms per mode independently of the size of the system. The computational cost of the X-TDH method is reduced significantly compared to the L-TDH method for systems with many operator terms per mode such as is typical for accurate molecular potential-energy surfaces. Numerical results for L-TDH and X-TDH are presented which confirm the theoretical reduction of the M scaling compared to standard first-quantization formulations. Calculations on Henon-Heiles potentials with more than 105 dimensions and polycyclic aromatic hydrocarbons with up to 264 modes have been performed. Thus, the SQ formulation and the X-TDH method pave the way for studying the time-resolved quantum dynamics of large molecules. Published by AIP Publishing. https://doi.org/10.1063/1.5049344

I. INTRODUCTION parts of building up general and efficient quantum-dynamical machinery. Many interesting phenomena in chemical physics are best Among theoretical methods proceeding beyond TDH, described using a time-dependent quantum-mechanical for- we mention in particular multi-configuration time-dependent mulation.1 The numerical calculation of accurate approximate Hartree (MCTDH),8,9 which is now a well-established and time-dependent wave functions is therefore an important the- versatile theory and computational method for calculating oretical challenge. For few-dimensional problems, there exist time-dependent wave functions. MCTDH combines the time- standard and sufficiently efficient approaches.1,2 However, for dependent variational principle (TDVP) with evolution of only slightly more degrees of freedom (more than four atoms), (now several) Hartree products and thus shares a number matters are complicated, and over the years, many methods of basic ideas and practical components with TDH, such as have been developed employing different ideas for how to mean-field matrices. Although MCTDH does not overcome parameterize wave functions and how to propagate these in the exponential growth of computational cost with increasing time. dimensionality, the method considerably alleviates the issue One of the basic methods for computing the time evolution by a variational optimization of the one-mode basis func- of a molecular system is the time-dependent Hartree (TDH) tions in a manner that can be systematically improved to the approach, also sometimes denoted as the time-dependent self- numerically exact limit. Still, the inherent exponential scal- consistent-field (TD-SCF). TDH is the time-dependent analog ing with the number of degrees of freedom is a potential of vibrational self-consistent-field (VSCF),3–7 which in turn is limitation, and several suggestions have been put forward to the analog of Hartree-Fock theory, only now for distinguish- mitigate this, such as multilayer MCTDH,10–12 or alternative able degrees of freedom corresponding to nuclear motion. The approaches tailored to specific types of complex systems such fundamental approximation of TDH is the adoption of a sin- as system-bath simulations.13,14 Multi-configurational expan- gle Hartree product as a wave-function ansatz which leads to sions have been combined with a time-dependent Gaussian a mean-field description of the interactions between the dif- basis or similar in various forms, such as the G-MCTDH15 ferent degrees of freedom. Being a mean-field approximation, and variational multi-configurational Gaussian (vMCG) (see TDH has significant shortcomings.2,8 Nevertheless, TDH and Ref. 16 for a review), and (two-layer) coupled-coherent states many of its theoretical and technical ingredients are essential (CCS).17 These methods are in turn somewhat related to other successful Gaussian-based approaches such as the multiple spawning methods; see Ref. 18 for a very recent review. A a)Present address: Dipartimento di Scienze e Innovazione Tecnologica, DiSIT, University del Piemonte Orientale “Amedeo Avogadro,” Alessandria, Italy. fundamental distinction is whether the basis has its own time b)Electronic mail: [email protected] evolution (i.e., classical Gaussians) or is described in terms

0021-9606/2018/149(13)/134110/16/$30.00 149, 134110-1 Published by AIP Publishing. 134110-2 Madsen et al. J. Chem. Phys. 149, 134110 (2018) of parameters determined by (variational) criteria applied to the elements of an anti-Hermitian matrix by ansatz preserves the full wave functions. Altogether there is still room for new orthogonality of the basis functions. A mathematical analysis ways of addressing the constituting elements of computational of the parameterization reveals how the actual transformations quantum dynamics. can efficiently be carried out and identifies periodicities and For time-independent wave functions, we have devel- thereby the relevant parameter ranges. The time-derivative oped a vibrational coupled-cluster (VCC) theory.19,20 The potentially results in quite involved equations for an expo- VCC method has proven to be an efficient method for nential parameterization, but from a mathematical analysis, it calculating anharmonic vibrational spectra and properties becomes clear that it is natural to consider the parameter space of molecules,21,22 and there has been significant progress in terms of projections on some specific directions. Employ- toward higher efficiency and larger molecular systems, e.g., ing these projections, very simple final equations are obtained by employing tensor decomposition methods.23 As we cur- for the time evolution of the X-TDH parameters. Similar to rently seek to generalize this approach to a time-dependent L-TDH, the X-TDH EOMs can be integrated without explic- framework, it is important to be able to describe the ref- itly solving linear equations and/or inverting matrices which erence state for the cluster expansion (a Hartree-product is a non-trivial result. In addition, X-TDH does not require state like VSCF/TDH) as well as general time-dependent the full mean-field matrix like L-TDH, offering significant transformations of the basis. computational savings. In this paper, we explore new ways of parameterizing TDH has the potential of being applicable to large sys- time-dependent wave functions and formulating equations for tems due to its low computational scaling, but realizing this the time evolution of quantum states. Specifically, we inves- potential for real molecules is a significant challenge. We have tigate the consequences of adopting an exponential parame- previously illustrated how VSCF can be applied to systems terization and second quantization (SQ) in conjunction with with thousands of degrees of freedom,6 and one of the goals the TDVP. Our formulation covers TDH, exact wave func- of the present formulation is to be able to carry over these tech- tions, and time-dependent vibrational configuration interac- nical advances to time-dependent wave functions. We illustrate tion (VCI). TDH is studied in significant detail for several rea- how the active-term algorithm of Ref.6 can be employed to sons, including (i) the general theoretical significance due to its provide highly efficient implementations of both L-TDH and status as a fundamental approximation in chemical physics, (ii) X-TDH. A low and predictable computational scaling with the its direct or indirect role in related theoretical frameworks such size of the system for Hamiltonians with a restricted level of as those mentioned above, and (iii) the use of TDH method- mode-coupling is obtained for both L-TDH and X-TDH and ology as a vehicle for pushing the size-limits for quantum illustrated in practice by applying the implementation to poly- simulations. cyclic aromatic hydrocarbons (PAHs) with up to 264 modes. In this work, we employ the previously introduced second- The efficiency gain of X-TDH compared to L-TDH is clearly quantization (SQ) formulation of many-mode dynamics.7 The seen for operators with many terms. Linear scaling is achieved SQ formulation requires, for a start, a significant amount of for both X-TDH and L-TDH in the limit where the number initial considerations and is in no way intended as a short of non-zero terms per mode is constant, making computations cut for TDH compared to existing first-quantization (FQ) for- with more than 105 modes possible. mulations (for FQ derivations of TDH, see Refs. 24 and 25). The paper is structured as follows. SectionII describes the However, once established, the SQ formulation provides a con- theoretical developments with Secs.IIA andIIB introducing venient framework supporting and inspiring alternative formu- the TDVP and the SQ formulation, respectively. In Sec. III, lations, implementations, and extensions. The SQ formulation a general TDH EOM is derived, which covers both linear has in this regard previously proved useful, for example, by and exponential parametrizations, and explicit EOMs for the automatically reducing the computational complexity by one L-TDH method are presented. In Sec.IV, EOMs for a general order in VSCF theory6 which is specifically interesting due to exponential parametrization are derived and concrete expres- the close relation between VSCF and TDH. We use the SQ sions for X-TDH are presented. SectionV presents EOMs for formulation to provide useful general relations for deriving an exponential parametrization of the exact method as well time-dependent equations and carrying out the time evolution as approximate time-dependent VCI. SectionVI describes the of operators and one-mode functions. We use these to formu- implementation, while our numerical results are presented in late operator-based equations for describing the time evolution Sec. VII. Finally, Sec. VIII provides a summary and future of the TDH state in terms of equations of motion (EOMs) outlook. for the SQ creation operators. With a linear parameterization of the time evolution of the SQ creation operators (or correspondingly the one-mode functions), these lead to the linear II. GENERAL THEORY TDH (L-TDH) equations requiring the full mean-field matrix A time-dependent wave function can be written in a phase- for their integration. We subsequently present a derivation separated form as and analysis of the EOMs for an exponentially parameterized TDH state, an approach we denote X-TDH, completing the Ψ¯ (q1, ... , qM , t) = exp(−iF(t))Ψ˜ (q1, ... , qM , t), (1) initial considerations of Ref. 26. Exponential parameteriza- tions have previously been highly useful in electronic structure where we consider a system with M degrees of freedom theory for non-redundant transformations of orbitals or other (modes) with coordinates {q1, q2, ..., qM }. The phase-isolated quantities.27 Expressing the unitary time evolution in terms of wave function, Ψ˜ , can be expressed as a linear combination 134110-3 Madsen et al. J. Chem. Phys. 149, 134110 (2018) of Hartree products, Φ˜ r, of one-mode functions (modals), The state with no occupation is denoted as the vacuum state, ˜m {φrm (qm, t)}, |vaci, which satisfies hvac|vaci = 1 and the killer condition, X m Ψ˜ (q1, ... , qM , t) = Cr(t)Φ˜ r(q1, ... , qM , t) arm |vaci = 0. (10) r We emphasize that this set of operators is fundamentally with different from the ladder operators used in the harmonic- YM oscillator problem, and the formulation is in no way restricted ˜ ˜m Φr(q1, ... , qM , t) = φrm (qm, t). (2) to harmonic-oscillator basis functions or reference states. m=1 Hartree-product states are in second quantization represented In the TDH approximation, a single Hartree product is used as by a string of one-mode creation operators applied to the a wave-function ansatz. The time dependence of the modals is vacuum state. Thus the TDH state can be written as not strictly necessary for multi-configurational wave functions, YM but it is introduced in, e.g., MCTDH in order to obtain a more ˜ m † |Φii = a˜im |vaci. (11) compact form of the wave function. m=1

A. The time-dependent variational principle Here the M-dimensional index vector i = (i1, i2, ..., iM ) defines which particular modal is occupied for each mode. Note that A variational solution to the time-dependent Schrodinger¨ we allow ourselves to use the same state labels in first and equation (TDSE) is obtained by applying the TDVP to the second quantization though two different formal representa- phase-separated wave function of (1). This results in EOMs tions are actually in play. In the following, we will omit the i for the generalized phase factor F(t) and the phase-isolated subscript from the TDH wave function. wave function. One formulation of the TDVP is the least-action The following convention is adopted in the labelling of 28–31 variational principle, modals: im is the reference modal for the mth mode; am, bm, m m m m m m t1 ∂ c , d denote the virtual modals; and p , q , r , s denote δS ≡ δ L dt = 0 with L ≡ hΨ¯ |(H − i )|Ψ¯ i, modals of unspecified occupancy. t0 ∂t (3) 1. A note on commutator relations where δΨ¯ (t0) = δΨ¯ (t1) = 0. For a phase-separated wave function as in Eq. (1), this results in the equations We may consider Eqs. (7)–(9) to be canonical commutator relations for creation and annihilation operators, or we ∂ F˙ = hΨ˜ |(H − i )|Ψ˜ i, (4) could say that the creation and annihilation operators are in ∂t some sense canonical since they are directly tied to the under- ∂ 0 = Re hδΨ˜ |(H − i )|Ψ˜ i, (5) lying modals. We will, however, encounter other creation and ∂t annihilation operators, such as similarity-transformed ones or where we have required normalization at all times, i.e., time derivatives of ones; in either case, we have the following δhΨ˜ |Ψ˜ i = 2Re hδΨ˜ |Ψ˜ i = 0. The above equations can equally important result. m well be derived from projecting the Schrodinger¨ equation on If any annihilation operatora ˜rm and creation operator allowed variations and requiring that if δΨ˜ is an allowed vari- m † a˜sm are ultimately a linear combination of the canonical ation so is iδΨ˜ . This results in the same equation for F˙ and annihilation and creation operators of (7), i.e., the alternative (but equivalent) expression for determining the X 32,33 m m time evolution of the wave-function parameters, a˜rm = cpm apm , (12) pm ⊥ ∂ X 0 = hδ Ψ˜ |(H − i )|Ψ˜ i, (6) m † m † ∂t a˜sm = dqm aqm , (13) qm using orthogonal variations hδ⊥Ψ˜ |Ψ˜ i = 0. then their commutator is a scalar operator (i.e., a scalar times B. Second-quantization formulation the identity operator) because of many-mode theory m m † X m m † [ãrm ,a ˜sm ] = cpm dqm [apm , aqm ], (14) In the second-quantization (SQ) formulation of Ref.7, pmqm vibrational wave functions and operators are described in terms of creation and annihilation operators. The one-mode creation which is a sum of scalars. m† operator, apm , creates occupation in the modal labeled with 2. Operators in second quantization pm for mode m. Likewise, the Hermitian-conjugate operator, the annihilation operator, removes the modal. The operators The creation and annihilation operators are used to express satisfy the commutator relations, all states as well as all other operators. For future reference, we introduce the one-mode shift operator, m m0 † [a m , a 0 ] = δ 0 δ m m0 , (7) r sm mm r s m m† m E m m = a m a m , (15) m m0 r s r s [arm , a m0 ] = 0, (8) s which shifts occupation from level sm to rm in mode m. The m † m0 † [a , a 0 ] = 0. (9) shift operators satisfy the commutator relation, rm sm 134110-4 Madsen et al. J. Chem. Phys. 149, 134110 (2018)

m m0 m m X X m m X X [E m m , E 0 0 ] = δ 0 δ m 0m E m 0m − δ m 0m E 0m m . (16) Ω Ω Ω m m E m m p q r0m s0m mm q r p s p s r q = 0 + r s r s + m m m m m m m r s m1

Ωm1m2...mn h m1 m2 mn |Ωm1m2...mn | m1 m2 mn i rm1 rm2 ...rmn sm1 sm2 ...smn = r r ... r s s ... s .

The exact vibrational Hamiltonian can be expressed by well as the approximate VCI case, as is briefly discussed in such an expansion. However, since the exact potential-energy Sec.V. surface (PES) of the system is not generally known, we often In Eq. (1), we separate out a total phase of the wave func- choose to represent the Hamiltonian in the computationally tion. Correspondingly, we shall later limit ourselves to special efficient sum-over-products (SOP) form unitary transformations of the phase-isolated wave function, m X Y mt which means that the K matrices are special unitary matrices H = ct h , (18) that have zero diagonal elements and unit determinants. The t m∈mt diagonal elements describe complex phase shifts, but we use where hmt is a one-mode operator on mode m. The SOP form the F factor in Eq. (1) for describing the overall phase of the has been shown to be computationally convenient, e.g., in the total wave function, thereby circumventing redundant descrip- cases of MCTDH,8 VCC,19,20,23 and VSCF.6 The one-mode tions of phase, meaning we have one overall phase factor as operators can be expressed in either the time-independent or opposed to a phase factor for each mode. the time-dependent basis (indicated by a tilde) as The unitary-transformation operators of Eq. (20) are used X X to rotate the creation and annihilation operators to a new mt mt m† m ˜ mt m† m h = hrmsm arm asm = hrmsm (t)ãrm (t)ãsm (t). (19) basis, rmsm rmsm κm m −κm κm m † −κm κm m −κm m † m e E m m e = e a e e a m e = a˜ a˜ m . (21) Here the time evolution of integrals is such as to counteract r s rm s rm s the evolution of the creation/annihilation operators, thus at all This defines the similarity-transformed operators, times reproducing the same operator. m † κm m † −κm X m † Km a˜rm ≡ e arm e = apm [e ]pmrm , (22) m 3. Unitary transformations in second quantization p m κm m −κm X −Km m ≡ m m Unitary transformations in the modal space can be a˜sm e asm e = [e ]s p apm , (23) m described using the one-mode shift operators in analogy p m with orbital transformations in electronic structure theory. where in the case of TDH, the K matrices contain the Operators of the form time-dependent wave-function parameters. The similarity- transformed creation and annihilation operators are used to X m X X m m represent operators and wave functions in the time-dependent U(κ) = exp(κ) = exp κ = exp κ m m E m m r s r s basis. In order to fulfill Eq. (19), the transformation of the m m rmsm * + * + integrals of the Hamiltonian operator must be defined as (20) X , - , - mt −Km mt Km h˜ m m = [e ] m m h m m [e ] m m . (24) represent modal transformations (see Ref.7 for a proof) r s r p p q q s pmqm and are unitary, provided the operators κm and the corre- m m sponding matrix representations on the modal spaces Km Since the eK and e−K matrices are unitary, they can alter- are anti-Hermitian. Unitary transformations can therefore be natively be written as Um and Um† and equations-of-motion m m m parametrized by the elements of the K matrices, where we can be derived for the U -matrix elements instead of the κrmsm note that each mode has its own transformation matrix and parameters. This approach is taken in Sec. III B, while the m there are no κ parameters referring to more than one mode at EOMs for the κrmsm parameters are derived in Sec.IV. a time. Consider now the set of Hartree-products that can be described in a given orthonormal basis. Any Hartree-product 4. The resolution of the identity in second quantization state can be obtained by applying an exponential operator as the one given in Eq. (20) to an arbitrary Hartree-product state. In principle, the SQ formalism as introduced so far allows In the case where there is only one mode, the TDH treatment for arbitrary (non-negative integral) occupancies in the same is exact and thus covers exact states. Furthermore, a formula- mode, even in the same modal, although these have no first- tion equivalent to this one-mode case is also applicable—using quantization counterparts. We will, however, limit ourselves state transfer operators—to the exact many-mode limit as to physically relevant occupancies, i.e., states that are at most 134110-5 Madsen et al. J. Chem. Phys. 149, 134110 (2018) singly occupied in any mode (but which can still be unoccupied we obtain an expression with the resolution of the identity and as well). get Under this assumption, it holds true that any two annihila- m m † X X Y m0 † Y m0 m tion operators of the same mode applied to any physical state P m = a a 0 |vacihvac| a 0 a m q qm rm rm q |αi give zero since at least one of them will operate directly m⊂ rm m0 ∈m m0 ∈m  | N * +{z * +} on the vacuum state as in (10),  =I   , - , - m m arm asm |αi = 0. (25) m † m = aqm aqm . (30) The commutation relation (7) reflects the orthonormality of m P m Defining the projector P = qm P m onto states that have m q the 1-mode states {|r i} since occupancy in mode m leads to the definition of the orthogonal- m m m m † m m † projection operator, hr |s i = hvac|arm asm |vaci = hvac|[arm , asm ]|vaci = δrmsm . m m m (26) Oqm = P − Pqm . (31)

The identity operator can therefore be resolved as the orthogonal projection operator onto all possible physical states III. SECOND-QUANTIZATION DERIVATION OF TDH obtained as a direct product of these 1-mode states, TDH is based on a single-conﬁguration wave-function X X ansatz, I = |vacihvac| + |rm0 ihrm0 | m M m0 r 0 Y |Φ¯ (t)i = exp(−iF(t))|Φ˜ (t)i = exp(−iF(t)) a˜ m†(t)|vaci, X X im + |rm0 rm1ihrm0 rm1 | + ··· m=1 m m m0

The orthogonal variation can be written as where {tact(m)} contains only terms that operate on mode m. Also note that ⊥ m m m m X m † m hvac|δ a˜ m = hvac|δa˜ m O m = hvac|δa˜ m a˜ m a˜ m , (38) i i i i a a m,˜i X m mt mt a Frmsm = z˜ hrmsm (46) ∈{ } using the orthogonal-projection operator of Eq. (31). Inserting t tact(m) (38) into (37) and using (33) gives depends on the time-dependentz ˜mt; i.e., the integrals of the m m † X m m † m mean-field operator always depend on the time-dependent ihvac|[δa˜ m , a˜˙ ]|vaci = hvac|[δa˜ m ,a ˜ ]ã m i im i am a basis. m a The analysis in Ref.6 shows that removing inactive terms Y m0 Y m00 † × a˜ 0 H a˜ 00 |vaci im im from the mean-field operator in VSCF corresponds to a reduc- m0,m m00 tion of the computational complexity with respect to the num- m m m † +g ˜ imim hvac|[δa˜im ,a ˜im ]|vaci. (39) ber of modes in the system (M) by one order. For example, if the Hamiltonian contains only two-mode couplings, the com- Using the fact that commutators between elementary operators putational scaling with the number of degrees of freedom is are scalars (see Sec. II B 1) results in the following EOM for reduced from M3 down to M2. In addition, it was found that for the time evolution of the occupied modals: large molecules, some interactions were so weak that screen- m m† X m m† m,˜i m m m† ing techniques could lower the computational scaling in VSCF i[δa˜ m , a˜˙ ] = [δa˜ m ,a ˜ ]F˜ +g ˜ m m [δa˜ m ,a ˜ ], (40) i im i am amim i i i im even further, eventually down to linear in the uncoupled case. am While this computational scaling may seem intuitive given the where the mean-field operator Fm,˜i is the VSCF Fock operator only linear increase in the number of parameters, it is diffi- (see Sec. III A). Note that the first term on the right-hand side of cult to achieve in an actual numerical implementation with m m Eq. (40) is zero for all parallel variations where δa˜im = k · a˜im , Hamiltonian operators containing non-trivial couplings. Nev- while the second term vanishes for orthogonal variations as ertheless, we shall continue to draw benefit from the VSCF defined in (36). experience in this work. It is important to utilize active terms The time evolution of the phase is determined by (4) which in order to obtain low M scaling. As discussed in Ref.6, the results in partitioning of the mean-field operator into active and inac- XM tive terms can also be realized in first quantization. We do not ˙ ˜ ˜ m F = hΦ|H|Φi − g˜ imim . (41) know if this has been done in other TDH implementations, m=1 but it is included in both the algorithms we implement in the following. A. The mean-field operator The mean-field operator is a one-mode operator which B. Linear parametrization can be expressed in both the time-dependent and the time- If we choose to expand the time-dependent modals in independent bases as an orthogonal, time-independent basis defined by the cre- m† ˜ X ˜ X ˜ m m,i ˜ m,i m † m m,i m † m ation and annihilation operators {aqm } and {apm } satisfying the F = Frmsm a˜rm a˜sm = Frmsm arm asm , (42) rmsm rmsm usual commutation relations, we define the L-TDH method by writing the time-dependent operators as where X m,˜i m m† m† m† m ˜ ˜ ˜ a˜ m (t) = a m C m m (t), (47) Frmsm = hΦ|[[ãrm , H],a ˜sm ]|Φi (43) r q q r qm and X m,˜i ˜ m m † ˜ m m m∗ Frmsm = hΦ|[[arm , H], asm ]|Φi. (44) a˜sm (t) = apm Cpmsm (t), (48) pm The ˜i superscript indicates that the integration over modes in m the mean-field-operator elements is carried out using the time- where the C matrices are the L-TDH time-dependent modal dependent wave function |Φ˜ i. coefficients which are chosen to be unitary at t = 0 in order to It is evident from (43) that terms in the Hamiltonian fulfill (34). which contain no reference to mode m do not contribute to the Inserting these definitions into (40) results in mean-field operator. This observation allows for introducing X X X an active-term algorithm as also used in VSCF,6 m∗ ˙ m m m† ˜ m,˜i 0 = δCpmim i Cqmim [apm , aqm ] − Famim m m m X Y 0 p q a ˜ m,˜i ˜ m ˜ m t m† ˜ * Frmsm = cthΦ|[[ãrm , h ],a ˜sm ]|Φi . 0 t m ∈mt X m m m† m X m m m† × C m ,m [a m , a m ] − g˜ m m C m m [a m , a m ] X m mt m† Y m0t q a p q i i q i p q = c hΦ˜ |[[ã m , h˜ ],a ˜ ] h˜ |Φ˜ i m m t r sm q q + t∈{t } m0 ∈m \m / act(m) t X X m∗ ˙ m m ˜ m,˜i m m = δC m m iC m m − Cpmam F m m − g˜ m m C m m -, (49) X Y 0 X p i p i a i i i p i m t mt mt mt m m = c h˜ 0 0 h˜ m m = z˜ h˜ m m , p a t im im r s r s 0 * + t∈{tact(m) } m ∈mt \m t∈{tact(m) } which for arbitrary variations of the Cm-matrix elements .* /+ , - (45) gives , - 134110-7 Madsen et al. J. Chem. Phys. 149, 134110 (2018)

m X m m,˜i m m m m ˙ ˜ (κ ) m = κ m m , (58) iCpmim = Cpmam Famim + Cpmim g˜ imim . (50) + a a i am together with the vector of de-excitation parameters, Note that the variations of the Cm-matrix elements are in principle arbitrary, but the constraint operator gm ensures that m m∗ (κ− )am = −κ m m , (59) the time evolution conserves the orthonormality of the one- a i mode functions. Thereby, the Cm matrices stay unitary at all and the full vectors of parameters for the mode m, times. κm Equation (50) contains reference to the expansion coeffi- m + m κ = , (60) cients of the virtual modals C m m which are simply defined as m p a κ− the orthogonal complement to the occupied modals. Instead * + of explicitly constructing the orthogonal complement, the and the corresponding time-derivatives,, - L-TDH EOMs can be written in terms of the time-independent basis as κ˙ m κ˙ m = + , (61) X m m m,˜i m,˜i m m κ˙ − iC˙ m m = F − δ m m F˜ − g˜ m m C m m p i pmqm p q imim i i q i * + qm X so that all the objects depending, on- κ can also be seen as eff,m m m = hpmqm Cqmim , (51) depending on the m κ vectors. qm The mode-m part of the κ operator is then written as written in the matrix form as κm = vm T κm = κm T vm, (62) m iC˙ = heff,mCm. (52) where vm is a vector of shift operators (or state-transfer oper- m,˜i Note that the F˜ m m matrix element of the mean-field operator ators in the VCI limit). For later reference, we also define the i i m m m (for any m) is equal to the energy hΦ˜ |H|Φ˜ i apart from inactive vector u = [κ , v ], terms not included in the mean-field operator. vm um The final working equations are obtained by choosing a vm ≡ + , um ≡ + ≡ [κm, vm], (63) m m m specific constraint operator g , and we mention a few choices v− u− here. The obvious choice gm = 0 results in the following * + * + equations of motion: with elements defined, - as , - m m m X m,˜i m,˜i m ≡ ˙ m m ˜ [v+ ]am Eamim , (64) iCpmim = Fpmqm − δp q Fimim Cqmim , (53) m m m q [v− ]am ≡ Eimam , (65) m m m m∗ m X m∗ m u m ≡ κ v m −κ E κ E F˙ = hΦ˜ |H|Φ˜ i. (54) [ + ]a [ ,[ + ]a ] = amim imim + bmim ambm , (66) bm m m,˜i Choosing g = F , we obtain instead m m m m m X m m [u− ]am ≡ [κ ,[v− ]am ] = −κamim Eimim + κbmim Ebmam . (67) X m ˙ m m,˜i m b iCpmim = Fpmqm Cqmim , (55) qm † T m † m T Note that v+ = v− and κ+ = −κ− which ensures that κ is F˙ = hΦ˜ |H|Φ˜ i(1 − M), (56) an anti-Hermitian operator. We define commutators between the operator vectors as eff,m m,˜i and thereby h = F . matrices with elements [a,b]ij = [ai, bj],

[vm, vm] = [vm, vm] = 0, (68) IV. TDH IN EXPONENTIAL PARAMETRIZATION + + − − A. Derivation of equations of motion [vm, vm] = Em − vm1, (69) We now derive the EOMs for an exponentially + − v 0 m m m m m parametrized state of the form m m |Φi with [Ev ]a b = Eambm and v0 = Eimim . Note that Ev = 0 and vm |Φi = |Φi. |Φ˜ i = exp(κ)|Φi, (57) 0 We now define the similarity-transformed time derivative, where |Φi is a time-independent reference state. EOMs can be κ ! 1 ∞ n derived from Eq. (40) using the time-dependent creation and − ∂e − X (−1) D = e κ = e sκ κ˙esκ ds = (κ )n κ˙, (70) annihilation operators defined in Eq. (22). However, in order ∂t (n + 1)! [ 0 n=0 to cover exact states as well as TDH, we derive EOMs for a general single-exponential ansatz. where we have used the relation for derivatives of exponential We adopt a notation similar to the standard notation in operators34 as well as the Baker-Campbell-Hausdorff (BCH) the response-function community and, excluding the redun- expansion and defined the nth nested commutator with the κ n 2 dant parameters, we define here the vectors of excitation operator as κ[ , e.g., κ[ Ω = [κ,[κ, Ω]]. The D operator parameters, can be partitioned into terms for each mode, 134110-8 Madsen et al. J. Chem. Phys. 149, 134110 (2018)

X κm ! X m −κm ∂e m b+ D = e = Dm, (71) b ≡ , (79) ∂t m m m b− * + and each one-mode operator can be split into even and odd with , - contributions, m ˜ m,˜i [b+ ]am = −iFamim , (80) ∞ X 1 m 2n m m m,˜i m ∗ Dm = (κ[ ) κ˙ − ˜ − (2n + 1)! [b− ]am = iFimam = [b+ ]am . (81) n=0 ∞ E X −1 For the values of κ where Dm is non-singular, Eq. (77) is + (κm)2n+1 κ˙m = DE + DO. (72) (2n + 2)! [ m m equivalent to solving the EOMs, n=0 DE κ˙ m = bm, (82) As shown later in Secs.IVC andIVD, the two contributions m E can be written as for each mode. When Dm becomes singular (which happens at a well-defined set of points described later), Eq. (77) is trivially DE = vm T DE κ˙ m, DO = um T DOκ˙ m, (73) m m m m fulfilled and Eq. (82) cannot be used as it stands. This, however, E O can easily be handled in practice as discussed further in Sec.VI, where Dm and Dm are Hermitian matrices. We shall see that only the even terms contribute to the TDH EOMs for the coef- and in SubsectionsIVB andIVC, the precise locations of the ficients, and, conversely, only the odd terms contribute to the singularities are described. phase equation. The explicit structure of the D matrices is derived in Equations of motion for the κ parameters are obtained Secs.IVC andIVD, but first the exponential transformation from the TDVP [Eq. (5)] using the following variation, is analyzed in more detail. obtained with analogous considerations for the derivatives with respect to parameters as carried out above for the B. Analysis of the exponential transformation time-derivatives: Consider the matrix representation of a κm operator in a κ ! single-mode modal basis, X X m ∂e |δΦ˜ i = δκ m m |Φi r s ∂κm m T m rmsm rmsm 01×1 κ− Km = m , (83) ! κ+ 0(Nm−1)×(Nm−1) X X m κ ∂D = δκ m m e |Φi * + r s ∂κ˙m where there are Nm basis functions for mode m and therefore m rmsm rmsm , - Nm − 1 parameters in the κm column vector and the whole X + κ m T E m T O m κ matrix is of dimensionality Nm × Nm. We now introduce = e v Dm + u Dm δκ |Φi = e δκ¯|Φi. (74) m q q R = |κm |2 = −κm T κm, (84) Here the δκ¯ operator has been implicitly defined from the m + − + { κm} m vectors of variations δ and the D matrices. This results in as well as γ± vectors defined as having norm one and direction m ! along the corresponding κ± vectors, † −κ ∂ κ 0 = Re hΦ|δκ¯ e H − i e |Φi m m m m ∂t κ+ = Rmγ+ , κ− = Rmγ− , (85) † −κ κ m m = Re hΦ|δκ¯ e He − iD |Φi and K = RmΓ with m T 01×1 γ = 1 hΦ|[δκ¯†, e−κ Heκ − iD]|Φi Γm − 2 = m . (86) γ+ 0(Nm−1)×(Nm−1) 1 X m † E m T † −κ κ * + = 2 δκ DmhΦ|[(v ) , e He − iD]|Φi, (75) This particular form of the matrix allows for an easy evalua- , - m tion of the matrix representation of the exponential operator where we have used the fact that the odd term in δκ¯ cancels out exp(κ) since the powers of the Γm matrix can be calculated as m m since it only contains Eimim and Eambm operators. For arbitrary a blockwise product, variations, we obtain − m m 2 1 01×(N −1) − m m (Γ ) = , (87) 0 = DE hΦ|[vm, e κ Heκ ]|Φi − ihΦ|[vm, DE ]|Φi , (76) m m T m s s m 0(Nm−1)×1 γ+ γ− m T m T m T * + where v = (v− v ). After a few manipulations, the final m T s + , 01×1 −γ− - EOMs are obtained as Γm 3 −Γm ( ) = m = . (88) −γ+ 0(Nm−1)×(Nm−1) DE S(DE κ˙ m − bm) = 0, (77) * + m m In general, one has , - where ( m 2n 1 for n = 0 m m 1 0 (K ) = n−1 2n m 2 (89) S ≡ hΦ|[v , v ]|Φi = (78) (−1) Rm (Γ ) for n > 0, s 0 −1 * + m 2n+1 n 2n+1 m and (K ) = (−1) Rm Γ . (90) , - 134110-9 Madsen et al. J. Chem. Phys. 149, 134110 (2018)

Separating even and odd terms in the series expansion of −1 2 m Y = , (100) exp(K ) and using standard Taylor series, we obtain 2 −1 m m m 2 * + exp(K ) = 1 + sin(Rm)Γ + (1 − cos(Rm))(Γ ) which allows us to rewrite C, m as - m 1 2 m 2 2 † = 1 + sinc(Rm)K + 2 sinc (Rm/2)(K ) , (91) Cm = Rm(XmYXm − 1). (101) − 2 † where we have used 1 cos(x) = 2 sin (x/2) as well as the The matrices Xm and Xm are pseudoinverses of each other unnormalized sinc function. It is evident that exp(Km) is peri- since they satisfy the four Penrose conditions: (i) AA+A = A; m odic in Rm, with period 2π, so that values of K corresponding (ii) A+AA+ = A+; (iii) (AA+)† = AA+; (iv) (A+A)† = A+A. m m † to the same Γ can give the same exponential. Note that the γ± (Here A+ is the pseudoinverse of A and A is its Hermitian m vectors are not defined when κ = 0, i.e., Rm = 0. In that case, transpose.35) We can then define the orthogonal projector however, exp(Km) = 1 in agreement with the corresponding † limit (Rm → 0) of Eq. (91). Pm = XmXm, (102) Using (91), the following expression for the time- dependent creation operators is obtained: on the column space, colsp(Xm), of Xm, as well as the orthogonal projector m † m † X m † m a˜ = cos(R )a + sinc(R ) a κ m m , (92) im m im m am a i − am Qm = 1 Pm, (103) ⊥ which relates the linear and exponential parametrizations. Note on its orthogonal complement, colsp(Xm) [or equivalently m † m † 1 † † that the weight of aim ina ˜im is zero for Rm = (n + 2 )π the projector onto null(Xm), the null space of Xm]. m † m † † anda ˜ m = ±a m for Rm = nπ. These are the points (except Due to the fact that XmXm = 12×2, any matrix M of the i i † E form XmM2×2Xm satisfies that Rm = 0) where the Dm matrix becomes singular as seen in SubsectionIVC. M = MPm = PmM = PmMPm, (104) E C. Analysis of the D operator while

In order to calculate the nested commutators of Eq. (70) 0 = MQm = QmM. (105) (n) m n m in a convenient way, we ﬁrst deﬁne Dm = (κ[ ) κ˙ and write Cm can therefore be decomposed into uncoupled terms for the ⊥ (0) m m T m colsp(Xm) and colsp(Xm) parts, respectively, Dm = κ˙ = v κ˙ , (93) (1) m (0) m m m T m 2 † Dm = [κ , Dm ] = [κ ,κ ˙ ] = u κ˙ . (94) Cm = Rm Xm(Y − 12×2)Xm − Qm 2 † Explicit evaluation of the commutators gives after some = −Rm PmXm(12×2 − Y)XmPm + Qm . (106) manipulations Due to the orthogonality of the projectors, arbitrary powers of m m T m m m T m T [κ , u ] = [κ ,[κ , v ]] = v Cm, (95) Cm can be written as

m m n n 2n n † where we have introduced the 2(N − 1) × 2(N − 1) matrix Cm = (−1) Rm PmXm(12×2 − Y) XmPm + Qm . (107) Cm as E Using Eqs. (72) and (97), the Dm can then be written as κmκm T −2κmκm T C = + − + + − R2 1. (96) m m m T m m T m X∞ X∞ −2κ− κ− κ− κ+ E 1 (2n) m T 1 n m Dm ≡ Dm = v Cm κ˙ * + (2n + 1)!  (2n + 1)!  Here 1 is the 2(Nm − 1) × 2(Nm − 1) identity matrix recalling n=0 n=0  , -   2(Nm − 1) as the number of κ coefﬁcients for mode m. = vm T DE κ˙ m,  (108) m   With (93) and (94) as the base cases and using (95) in the inductive step, it is seen by induction that we generally have, with for all n ∈ N0, ∞ n E X (−1) 2n n † D = P X R (1 × − Y) X P (2n) m T n m m m m (2n + 1)! m 2 2 m m Dm = v Cmκ˙ , (97) n=0    (2n+1) m T n m ∞  n  Dm = u Cmκ˙ . (98) X (−1)  +  R2n Q . (109) (2n + 1)! m m When R ≡ kκm k 0, γm ≡ κm/R is well deﬁned, and n=0  m ± , ± ± m   we can thus proceed further by introducing the 2(Nm − 1) × 2   The matrix   dimensional Xm matrix 2 −2 m 12×2 − Y = (110) γ+ 0 −2 2 Xm = m (99) 0 γ− * + − † * + is unitarily diagonalizable, 12,×2 Y -= U diag(4, 0)U , and the 2 × 2 matrix (common to all modes) where , - 134110-10 Madsen et al. J. Chem. Phys. 149, 134110 (2018)

1 1 1 DO = um T DOκ˙ m, (119) U = √ . (111) m m 2 −1 1 * + where n The required powers of the matrix are thus (12×2 − Y) ∞ 2n † , - X −1 = U diag(2 , δ0n)U , and recognizing that the infinite series DO = Cn . (120) m (2n + 2)! m correspond to those of unnormalized sinc functions [i.e., n=0 sinc(x) ≡ sin(x)/x for x , 0], we obtain the result Calculations similar to those in Sec.IVC give sinc(2R ) 0 E m † † ! Dm = PmXmU U XmPm + sinc(Rm)Qm. − 1 sinc2(R ) 0 0 1 DO = P X U 2 m U†X† P − 1 sinc2( 1 R )Q . m m m − 1 m m 2 2 m m * + 0 2 (112) , - (121) This expression for DE has been derived under the assumption m As for the DE matrix, DO is ill defined for R = 0, but we that R 0. We see that since lim sinc(x) = 1, m m m m , x→0 O − 1 see that limRm→0 Dm = 2 1. Defining the limit to be its value E † † for R = 0 makes DO a continuous matrix function of κm. lim Dm = PmXmU12×2U XmPm + Qm m m Rm→0 O Furthermore, this is consistent with the definition of the Dm m m = Pm + Qm = 1, (113) operator, which equals zero when κ = 0 (and thus u = 0). O The Dm matrix elements that contribute to the time regardless of the limit limRm→0 Xm. Furthermore, notice that evolution of F can then be evaluated as for R = 0, i.e., κm = 0, the DE operator is equal to its first m m term, DE = κ˙m = vm T κ˙ m. Thus, defining DE to be equal to its O m m T m O m m T m T O m m m hΦ|Dm |Φi = hi |u |i iDmκ˙ = κ− −κ+ Dmκ˙ limiting value 1 for R = 0 means that DE is continuous as a m m m † m † O m m E = − κ −κ D κ˙ (122) function of κ and consistent with Dm, also for Rm = 0. + − m We can rewrite the projector in terms of two projections † † † † m γm − γm † γm γm † by using the expressions for [u± ]am in (66) and (67) and noting on vectors ( + − ) and ( + − ) , m that only Eimim terms contribute. † † † Pm = XmXm = XmU12×2U Xm = Pm,1 + Pm,2, (114) Observing that ( with κm † −κm † for k = 1 κm † −κm † P = + − (123) + − m,k 0 for k = 2, 1 0 1 γm P X U U†X† + m,† − m,† m,1 = m m = m γ+ γ− , 0 0 2 −γ− the only contribution stems from the Pm,1 term, yielding the * + * + (115) following form of the phase equation: ,0 0 - ,γm - † † 1 + m,† m,† X Y mt Pm,2 = XmU U X = γ γ . (116) F˙ = hΦ˜ |H|Φ˜ i − ihΦ|D|Φi = c h˜ m m 0 1 m 2 γm + − t i i − t m * + * + E m This means that we can obtain the D in a simple form as X m † m † κ˙ + , - , m - − i − 1 sinc2(R ) (−1) κ − κ . (124) 2 m + − κ˙ m E m − Dm = sinc(2Rm)Pm,1 + Pm,2 + sinc(Rm)(1 − Pm,1 − Pm,2). * + (117) , - E. The projected κ˙ m equations E It is clearly seen from the definition of the projectors that Dm E When the Dm matrices are non-singular, the TDH state is Hermitian. The structure of the matrix is can be propagated by solving Eqs. (82) and (124). In the limit m AB K → 0, Eq. (82) reduces to DE = , (118) m B∗ A∗ κ˙ m = bm(κ = 0). (125) * + where A = A† and B = BT . This structure is similar to the E[2] , - We note that diagonalization of the mean-field operators matrix arising in VSCF and VCI response theory.36 corresponds to a stationarity condition as expected. If the As stated in Sec.IVB, DE is singular for R ∈ π \{0}. m m 2 N ground state is optimal relative to the Hamiltonian, we However, its reduction to the identity matrix for Rm → 0 means ˜ have a VSCF state satisfying the Brillouin theorem, F˜ m,i that it is always possible to change the modal basis in order amim m m = hΦ˜ |[E˜ m , H]|Φ˜ i = 0 and thus κ˙ = 0 and F(t) = hΦ˜ |H|Φ˜ it, to obtain κ = 0 when the Rm related to the previous κ imam approach π/2, and use the κm → 0 limit [Eq. (125)] to carry which is the expected result. on the first step after the change of basis. Equation (82) can be solved numerically. As it stands, it E appears that Dm must be handled explicitly. However, from the preceding analysis, one can derive simple and concise expres- D. Calculation of DO sions using projection techniques. Accordingly we separate O m Using Eqs. (98) and (72), Dm can be rewritten as out κ˙ into components 134110-11 Madsen et al. J. Chem. Phys. 149, 134110 (2018)

m m m m m κ˙ = Pm,1κ˙ + Pm,2κ˙ + (1 − Pm,1 − Pm,2)κ˙ independent of limRm→0 γ+ , m m m = κ˙ P + κ˙ P + κ˙ Q. (126) m = < m m m 1 2 lim κ˙ + = iym + ym − ym γ + b+ = b+ , (134) → + Rm 0 |f {z }g Projecting now (82) with Pm,1, Pm,2, and Qm, using (117), and =0 m m noticing that by the anti-Hermitian relations for γ± and b± , lim F˙ = hΦ˜ | H | Φ ˜ i, (135) Rm→0 m 1 m † m † b+ 1 m † m m † m ∗ in agreement with (125). γ+ ∓γ− m = γ+ b+ ∓ (γ+ b+ ) 2 b− 2 Since the X-TDH working equations have singularities for m * + = some κ values (and thus some values of Rm), the possibility iy (−; Pm,1) , - = m (127) of the absence of global solutions has to be considered. A y< (+; P ),  m m,2 simple procedure (described in Sec.VI) can be implemented  allowing the singularities to be avoided while propagating the where we have defined   modals. < = m † m The final working equations for the L-TDH and X-TDH ym = y + iy = γ b , (128) m m + + methods are summarized in TableI. we obtain F. Calculation of the transformed integrals m m m = γ+ sinc(2Rm)κ˙ = Pm,1b = iy , (129) In order to construct the occupied-occupied and virtual- P1 m −γm − occupied blocks of the time-dependent one-mode inte- * m + m m < γ+ grals that are needed for expectation values and mean- κ˙ = Pm,2b = y , ,- (130) P2 m γm field operators, we define the half-transformed matrix − mt P mt Km hˇ = m h [e ] m m . The time-dependent integrals are * + m rmsm p rmpm p s m m m ym γ+ mt P −Km mt − then obtained as h˜ m m = m [e ] m m hˇ m m . Using Eq. (91), sinc(Rm)κ˙ Q = Qmb = b , ∗- m . (131) r s p r p p s ym γ− we arrive at the expressions * + κ˙ ˜ mt ˇ mt m We see that the structure of preserves, the anti-Hermitian- himim = cos(Rm)himim + sinc(Rm)Iimim , (136) property as it should. All in all, we can propagate the X-TDH ˜ mt ˇ mt m ˇ mt state by calculating hamim = hamim − sinc(Rm)κamim himim

= < 1 2 m m 2iy y y 1 − sinc (Rm/2)κ m m I m m , (137) κ˙ m = m + m − m κm + bm 2 a i i i + sin(2R ) R sin(R ) + sinc(R ) +  m m m  m using the intermediates  =  X  1 iy  ˇ mt mt mt m  m  < m m h m m = cos(Rm)h m m + sinc(Rm) h m m κ m m , (138) =  + sinc(Rm)ym − ym κ+ + Rmb+ . i i i i i b b i sin(R ) cos(R ) bm m  m  * +  X  (132) mt mt mt m hˇ m m = cos(R )h m m + sinc(R ) h κ , (139) , -  a i m a i m ambm bmim bm The corresponding negative-index part can be calculated by X m m∗ ˇ mt complex conjugation, and in this manner, we only need to Iimim = κamim hamim . (140) calculate the positive-index part of the mean-ﬁeld vector bm. am The EOM for the phase factor (124) can be rewritten as G. Auto-correlation functions in X-TDH X Y X ˙ ˜ mt = F = ct himim + tan(Rm)ym, (133) The time-dependent wave function can be used as a tool t m m for calculating spectra by introducing the auto-correlation function S(t),1,8,37 which for X-TDH takes the form [using by using (129). Note that F˙ is real in accord with the Eq. (92)] requirements set up earlier. −iF(t) Y As has been seen before, (132) and also (133) appear S(t) = hΦ¯ (0)|Φ¯ (t)i = e cos(Rm). (141) to be ill deﬁned for Rm = 0, but fortunately the limits are, m

TABLE I. Working equations for the L-TDH and X-TDH methods. Note that the L-TDH equations depend on the choice of the gm operator.

L-TDH X-TDH

= m X m,˜i m,˜i m m m 1 iym < m m iC˙ = F − δ m m F˜ − g˜ C κ˙ = + sinc(R )y − y κ + R b pmim pmqm p q imim imim qmim + sin(R ) cos(R ) m m m + m + qm m  m  * +  X Y XM  X Y X  ˙ ˜ mt m ˙  ˜ mt =  F = ct himim − g˜imim F =, ct himim + tan(Rm)y-m  t m m=1 t m m 134110-12 Madsen et al. J. Chem. Phys. 149, 134110 (2018)

If the initial wave function is real and the Hamiltonian is orthogonal complement indexed as |Ai, we can write states symmetric, i.e., H = HT , the following expression can be (VCI or exact or other) as used for calculating the auto-correlation function at longer X times: ˜ ∗ |Φi = exp (KA(t)EAI − KA(t)EIA) |Ii. (146) ∗ A S(2t) = hΦ¯ (t)|Φ¯ (t)i * + Following the above, derivation, the time evolution- is given by −2iF(t) Y 2 2 X m 2 = e cos (Rm(t)) + sinc (Rm(t)) (κ m m (t)) . ! a i 1 iY = m  am  K˙ R Y < − Y K R B   + = + sinc( ) + + + .  (142) sin(R) " cos(R) #   (147) The absorption cross section at photon energy E can then be Here we have separated out the norm of the parameters as obtained as the Fourier transform of S(t),8,37 previously, i.e.,

∞ R = |K+|. (148) σ(E) ∼ E S(t)ei(Ei+E)tdt, (143) −∞ The number Y is obtained from 1 † where Ei is the energy of the initial wave packet. Y = Y < + iY = = K B , (149) R + +

H. Computational scaling of X-TDH and L-TDH in analogy to Eq. (128). The elements of the B+ vector are obtained from the Hamiltonian matrix elements, Comparing the X-TDH working equations [Eq. (132)] to the L-TDH counterpart (51) shows that in each derivative eval- [B+]A = −ihA˜ |H|I˜i = −iH˜ AI . (150) uation the L-TDH method requires the construction of the full active-term mean-field matrix of size N × N, where N The tilde denotes that we are using the transformed-state basis. is the number of modals per mode (we here assume Nm = This can be related back to the integrals of the Hamiltonian in N ∀ m). X-TDH only needs the virtual-occupied block (one whatever primitive state basis is used. These relations follow m column vector of size N − 1) which goes into the b+ vector. the transformations in SubsectionIVF, only the transfor- The X-TDH method thereby saves a factor N in constructing mation is in state space instead of modal space. Altogether, the mean field which is the primary bottleneck for systems the exact-state parameters can be evolved using the evolu- with many operator terms per mode (such as molecular PESs). tion of the unitary configuration-space parameters as given When the operator contains few terms per mode or N is very by Eq. (147). The right-hand side of this equation can be eval- large, the bottleneck is instead matrix-vector products in the uated based on transformed Hamiltonian integrals. Inspecting integral transformation scaling as O(N2) for both methods. the details, it is seen that the state can be propagated based on transformations of the Hamiltonian matrix followed by some dot products and vector additions. In this work, we focus on the V. EXPONENTIAL PARAMETRIZATION OF EXACT efficiency gains of the exponential parameterization for TDH. AND VCI WAVE FUNCTIONS Whether there are any gains of the formulation for exact or The exponential parameterization also covers the cases of truncated time-dependent VCI calculations is a more compli- an exact wave function and a truncated VCI wave function cated issue and will be investigated in the context of future by a simple specialization. In this case, we remove the mode work. We therefore do not present here the further details index and consider the shift operators EPQ as state-transfer specializing the Hamiltonian matrix transformations, but all operators, i.e., the operator ingredients are available. We note that the presented single-exponential parameter- EPQ = |PihQ| (144) ization presents an alternative to the standard linearly parameterized method for time evolution of any normalized state in an transfers the system from state |Qi to |Pi. In addition to remov- orthonormal basis. The type of parameterization is also directly ing the m index, we use capitalized letters to index the state relevant for a potential double-exponential parametrization of space in a manner distinguishable from the one-mode analogs. MCTDH, which is a topic for future research. Note that these state-transfer operators satisfy commutator relations corresponding to the commutator relations of the one-mode shift operator in Eq. (145), VI. IMPLEMENTATION SCHEME A wave-packet-propagation algorithm based on the TDH [EPQ, ERS] = δQREPS − δPSERQ. (145) theory developed in Secs.II–IV is implemented in the In this manner, the unitary transformations introduced in MidasCpp38 program package. The basic algorithms of the Sec. II B 3 are unitary transformations in the state space. time-independent code can be reused, e.g., integral calculation Because the parameterization and commutator relations are and the active-terms algorithm for constructing the mean-field equivalent, the same derivation applies, and the final result is operator.6 The wave packet is represented in terms of a primi- essentially obtained by removing the m index. tive time-independent basis which can be harmonic-oscillator Thus, if we denote the reference state for the time evo- functions, distributed Gaussians,39,40 or B-splines.41 Based on lution as |Ii with an orthonormal set of basis states for its this primitive basis, an orthonormal basis is constructed in 134110-13 Madsen et al. J. Chem. Phys. 149, 134110 (2018) some way (e.g., using VSCF modals) and used in the TDH VII. RESULTS calculation. A. Computational details The X-TDH and L-TDH implementations can be efficiently parallelized over both modes [using Message Passing In the following, numerical results of the L-TDH and Interface (MPI)] and operator terms (using OpenMP). This has X-TDH methods are presented. Calculations are performed on been done in both the integral transformations and the deriva- the set of polycyclic aromatic hydrocarbons (PAHs) of Ref.6 tive calculation where the loop over modes is parallelized which include up to two-mode couplings in the PESs and on using MPI and the loop over active terms in the mean-field high-dimensional Henon-Heiles systems. construction is threaded using OpenMP. As a primitive basis, we use B-splines for all calculations. For integrating the EOMs, we use our own implementa- The initial wave packet is generated by performing a VSCF tion of the Dormand-Prince 8(5,3) (DOP853) explicit Runge- calculation on a second PES using the same primitive basis Kutta method42 with adaptive step-size control which also set. In this way, the vibrational wave function of one PES is contains a high-order interpolant for dense output. The dense- placed on another potential on which the time propagation is output feature is needed in order to obtain auto-correlation performed. In order to obtain an orthonormal modal basis in functions at equidistant time points which is required for which the reference state is contained, we transform from the using the Fast Fourier Transform (FFT) for calculating spec- primitive basis to the spectral basis of the VSCF calculation tra. We have also implemented and tested the Dormand-Prince before starting the TDH propagation. This also allows us to 5(4) (DOPRI5) method,42 but in our experience, the DOP853 truncate the basis based on the VSCF modal energies which method performs much better in terms of both speed and may be done when the two PESs are similar and the wave accuracy. It should also be noted that the step sizes used for packet does not change dramatically over time. integrating the L-TDH and X-TDH EOMs to a given accuracy All results (except the ones in Subsection VII E) are are very similar in all test cases. obtained using the serial version of the TDH implementation. The singularities in the X-TDH EOMs are avoided by using the simple procedure described in the following. B. Dynamics of tetracene 1. Define a threshold T ∈ [0, 1]. R As a first test of the X-TDH method, we study the tetracene 2. If at some point, R > T × π . m R 2 molecule (denoted as PAH4 in Ref.6) with 84 vibrational (a) As a first attempt, reduce the step size by a factor of modes and a PES including up to two-mode-coupling terms. two and retry the step. The initial wave packet was generated by removing all cou- (b) If this fails, transform the primitive integrals of mode plings and qn terms with n > 2 from the PES and perform- m using the last accepted step and reset κm to zero, mt mt m ing VSCF on this potential. Propagation was performed for i.e., h m m ← h˜ m m and κ = 0, before retrying the 6 m r s r s tend = 10 a.u. ≈ 24.2 ps using the N = 48 lowest VSCF modals rejected step. as basis for each mode, and the resulting auto-correlation func- The kappa vectors that have been used to transform the tion is shown in Fig.1 [note that we can calculate S(t) at integrals need to be saved in order to calculate auto- twice the time using Eq. (142)]. The auto-correlation function correlation functions for later times. If at time t the kappa ∗ ∗ ∗ shows a significant structure. Because the PES is not har- vector has been reset at times t1 < t2 < . . . < tn, ∗ monic, no complete revivals are observed in the studied time the auto-correlation functions at t > tn are calculated interval. from the set of saved kappa vectors κ(t∗), κ(t∗), ... , κ(t∗) 1 2 n The spectrum obtained as the Fourier transform of S(t) and the final kappa vector κ(t) by defining the total is shown in Fig.2. As expected, the most intense peak cor- unitary-transformation matrix Um(t), responds to the VSCF ground-state energy (EVSCF = 2.397 × 10−1 a.u.) and a few small peaks are observed at higher m † X m † X m ∗ X m ∗ K (t1) K (t2) a˜ m (t) = a m [e ] m m [e ] m m ··· energies. The intensities of the excited-state peaks are weak i r r s1 s1 s2 rm sm sm because the initial wave packet is quite similar to the VSCF * 1 2 . state. X X ,Km(t) m † m × [e ] m m = a m U m m (t). (151) sn+1i r r i m rm sn+1 /+ The auto-correlation function- is then calculated as M −iF(t) Y m S(t) = e Uimim (t), (152) m=1 or under the conditions described above Eq. (142),

M −2iF(t) Y X m 2 S(2t) = e (Urmim (t)) . (153) m=1 rm

Note that Eqs. (152) and (153) reduce to Eqs. (141) and FIG. 1. Absolute value of the auto-correlation function of propagating (142) when there are no reset kappa vectors. tetracene from an initially Gaussian wave packet. 134110-14 Madsen et al. J. Chem. Phys. 149, 134110 (2018)

FIG. 2. Normalized spectrum of tetracene obtained from the Fourier transform of Fig.1. The blue line in the middle of the most intense peak is the VSCF ground-state energy. Note that the top of the most intense peak (which is normalized to 1) is not shown in order to see the less intense features of the spectrum.

C. Comparison of the L-TDH and X-TDH methods The L-TDH and X-TDH methods are two different parametrizations of the same wave function, and thus the results obtained from the two methods are the same. As dis- FIG. 3. M scaling of the cost of a single step in propagating the L-TDH and cussed in Sec.IVH, the X-TDH method only requires one X-TDH EOMs for an array of PAHs using different numbers of modals. (a) column of the mean-field matrix, while the L-TDH method Nm = 32 and (b) Nm = 64. requires the whole matrix. This saves a factor of N in the mean-field calculation and thus reduces the computational cost M 2 M M−1 ! 1 X ∂ 1 X 2 X 2 1 3 of propagating the TDH wave function significantly whenever H = − + qm + λ qmqm+1 − q , 2 ∂q2 2 3 m+1 the mean-field calculation is the primary bottleneck. This is m=1 m m=1 m=1 the case when the PES contains many terms per mode which (154) is likely to be the typical case and certainly true for the PAH with λ = 0.111 803. Since only nearest-neighbor couplings are examples considered here. included in the potential, the theoretical M scaling of our TDH Figure3 shows the average cost of a single propaga- methods is linear. Note that this scaling is not trivial to obtain in tion step for X-TDH and L-TDH calculations on PAHs. For practice for large systems. A series of calculations on systems L-TDH, gm = 0 is used (in our experience, changing the con- with different numbers of dimensions have been performed straint operator has very little impact on the step sizes in the where the initial wave packets were constructed as the ground- integrator). The initial wave packets are generated using the state wave function of a shifted harmonic oscillator with same procedure as in Sec. VII B and propagated to t = end V = 1 PM (q − 1)2. Note that because we have not imple- 103 a.u. Note that the number of propagation steps is approx- 2 m=1 m mented a complex absorbing potential (CAP) in MidasCpp, imately the same for both methods and thus the time per we had to use initial conditions different from the ones of step reflects the total computation time. The data for each Refs. 43 and 44 and thus the results are not easily comparable. method have been fitted to a power function f (M) = BMa, Figure4 shows the cost per step of the X-TDH method for and in both cases, the scaling is close to the theoretical value calculations with up to 165 000 dimensions, and it is clear O(M2). The observed M scaling is slightly lower for X-TDH that the scaling is very close to linear. The Henon-Heiles because the cheaper mean-field calculation does not com- potential is a simple model system, but linear scaling of our pletely overshadow the cost of the integral transformations TDH methods will also be observed for more complicated (which scale linearly with M) for the smaller molecules. It is clear from Fig.3 that the X-TDH method is much faster than L-TDH. The gain of using X-TDH increases with M because of the increasing number of mode couplings for larger molecules. For larger basis sets, the advantage of X-TDH is also larger but not twice as big for Nm = 64 compared to Nm = 32 because the cost of the integral transformations also increases.

D. High-dimensional Henon-Heiles potentials In order to study the behavior for very large systems, we perform X-TDH calculations on the high-dimensional Henon- FIG. 4. M scaling of the cost of a single step in propagating the X-TDH EOMs Heiles potentials described in, e.g., Refs. 43 and 44, for high-dimensional Henon-Heiles potentials. 134110-15 Madsen et al. J. Chem. Phys. 149, 134110 (2018)

2 FIG. 5. Spectra of 10D, 100D, and 1000D Henon-Heiles potentials from propagation of a shifted Gaussian wave packet to tend = 5 × 10 a.u. using X-TDH. systems as long as the number of operator terms per mode is law, constant. 1 S(s) = p , (155) In Fig.5, spectra of the 10D, 100D, and 1000D Henon- (1 − p) + s Heiles systems are shown. As expected, the number of peaks where s is chosen to be the number of parallel processes in (and thereby the number of states included in the initial wave order to obtain an estimate for the parallel proportion of the packet) increases with the system size. execution time, p. In both cases, p > 90%. We can thus conclude that significant speedups can be obtained by parallelizing over either modes or operator terms. E. Parallelization Which one is more favorable will depend on the calculation. Figure6 shows the parallel speedup in an X-TDH cal- For systems with many operator terms per mode (like the 3 culation on PAH8 (120 modes) with N = 48 and tend = 10 PAHs), approximately the same speedup can be obtained from a.u. using different numbers of threads for parallelizing over both schemes. On the other hand, for high-dimensional sys- operator terms in the integral transformation and mean-field tems with few coupling terms (like the Henon-Heiles systems), construction. The figure clearly shows that the wall time can parallelization over modes will be most advantageous. The be reduced significantly by using more CPUs. In Fig.7, the two levels of parallelism are also easily combined allowing parallel speedup is shown for using different numbers of MPI for large-scale calculations on multiple computer nodes using processes in the loops over modes in the integral transforma- both MPI and OpenMP. To give an absolute timing, on aver- tion and derivative calculation. Also here, a significant parallel age one-time step (of average length ∼3 a.u.) in the 264-mode speedup is possible. Both data sets have been fitted to Amdahl’s 2.29 × 106-term PAH24 X-TDH computation takes 1.4 s

FIG. 6. Parallel speedup in integrating the X-TDH equations for PAH8 (120 modes) using different numbers of OpenMP threads in the integral transformations and mean-ﬁeld construction. The data have been ﬁtted to Amdahl’s law.

FIG. 7. Parallel speedup in integrating the X-TDH equations for PAH8 (120 modes) using different numbers of MPI processes in the integral transformations and derivative evaluation. The data have been fitted to Amdahl’s law. 134110-16 Madsen et al. J. Chem. Phys. 149, 134110 (2018) with 12 MPI processes distributed over 3 nodes with the 7 5J. M. Bowman, Acc. Chem. Res. 19(7), 202 (1986). OpenMP threads per MPI process. Note that the speedup from 6M. Hansen, M. Sparta, P. Seidler, D. Toffoli, and O. Christiansen, J. Chem. parallelizing over multiple nodes depends on the performance Theory Comput. 6(1), 235 (2010). 7O. Christiansen, J. Chem. Phys. 120, 2140 (2004). of the network. 8M. Beck, A. Jackle, G. Worth, and H. Meyer, Phys. Rep. 324, 1 (2000). 9H.-D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett. 165(1), 73 (1990). VIII. SUMMARY AND OUTLOOK 10H. Wang and M. Thoss, J. Chem. Phys. 119, 1289 (2003). 11U. Manthe, J. Chem. Phys. 128, 164116 (2008). We have introduced a new second-quantization formu- 12H. Wang, J. Phys. Chem. A 119(29), 7951 (2015). lation of TDH and derived equations of motion for both a 13I. Burghardt, M. Nest, and G. A. Worth, J. Chem. Phys. 119(11), 5364 linear (L-TDH) and an exponential (X-TDH) parametriza- (2003). 14 tion of the unitary modal transformations. The formulation M. Bonfanti, G. F. Tantardini, K. H. Hughes, R. Martinazzo, and I. Burghardt, J. Phys. Chem. A 116(46), 11406 (2012). also covers exact states as well as approximate VCI wave 15I. Burghardt, H. Meyer, and L. Cederbaum, J. Chem. Phys. 111, 2927 functions when the one-mode shift operators are replaced by (1999). state-transfer operators. The X-TDH method is significantly 16G. Richings, I. Polyak, K. Spinlove, G. Worth, I. Burghardt, and B. Lasorne, faster than L-TDH because it only requires one column of the Int. Rev. Phys. Chem. 34(2), 269 (2015). 17J. A. Green, A. Grigolo, M. Ronto, and D. V. Shalashilin, J. Chem. Phys. mean-field matrix in order to propagate the EOMs. Numerical 144(2), 024111 (2016). results which confirm the computational benefits of X-TDH 18B. F. E. Curchod and T. J. Mart´ınez, Chem. Rev. 118(7), 3305 (2018). have been presented for a set of PAHswith up to 264 modes and 19O. Christiansen, J. Chem. Phys. 120(5), 2149 (2004). 20 high-dimensional Henon-Heiles systems with more than 105 P. Seidler and O. Christiansen, J. Chem. Phys. 131(23), 234109 (2009). 21B. Thomsen, M. B. Hansen, P. Seidler, and O. Christiansen, J. Chem. Phys. modes. Thus, the X-TDH method paves the way for studying 136(12), 124101 (2012). the quantum dynamics of very large systems. 22I. H. Godtliebsen and O. Christiansen, J. Chem. Phys. 143(13), 134108 The theoretical analysis of the exponential parametriza- (2015). 23N. K. Madsen, I. H. Godtliebsen, S. A. Losilla, and O. Christiansen, J. 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