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Topologically Correct Quantum Nonadiabatic Formalism for On-The-Fly Dynamics Loïc Joubert-Doriol, Janakan Sivasubramanium, Ilya G. Ryabinkin, and Artur F. Izmaylov J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b02660 • Publication Date (Web): 30 Dec 2016 Downloaded from http://pubs.acs.org on January 6, 2017

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The Journal of Physical Chemistry Letters

Topologically Correct Quantum Nonadiabatic Formalism for On-the-fly Dynamics Lo¨ıc Joubert-Doriol,†,‡ Janakan Sivasubramanium,† Ilya G. Ryabinkin,†,‡ and Artur F. Izmaylov∗,†,‡ †Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario, M1C 1A4, Canada ‡Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada E-mail: [email protected]

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The time-dependent variational principle (TDVP) 1–3 provides a very efficient framework for simulating quantum dynamics in large molecular systems. The most powerful aspect of this framework is use of time-dependent basis functions which reduces basis set size requirement compare to that for static basis sets. Two most widely used branches of the TDVP methodology constitute approaches related to the multi-configuration time-dependent Hartree (MCTDH) method 4–6 and approaches using frozen-width Gaussian functions. 7–15 If the MCTDH-based approaches are more suitable for fixed diabatic models, the frozen Gaussian functions have been extended to simulating nuclear dynamics with the on-the-fly calculation of the electronic potential energy surfaces. 7–9,16 Naturally, the adiabatic representation becomes the most straightforward representation for the electronic part of the problem in this case. One of the most frequent manifestations of the nuclear quantum character is nonadiabatic phenomena where the nuclear dynamics involves several electronic states. TDVP has been successfully extended and applied to modelling nonadiabatic dynamics (NAD). Often NAD becomes necessary because adiabatic electronic potential energy surfaces form conical intersections (CIs). 17–20 CIs promote transitions between electronic states and introduce nontrivial geometric phases (GPs) 21–26 that can affect dynamics in profound ways. 27–32 It is important to stress that CIs and associated GPs appear only when one uses the adiabatic representation for description of electronic part of the total wave-function. CIs and GPs disappear when the diabatic representation is used, however, physical observables of NAD do not change with the representation. Therefore, the dynamical features that emerge in the adiabatic representation due to a nontrivial GP appear in the diabatic or any other representation as well. 33,34 One of the simplest signatures of the nontrivial GP introduced by CI is a nodal line appearing in non-stationary nuclear density that moves between minima of a double-well potential with a CI in between the minima (Fig. 1). This nodal line appears due to acquisition of opposite GPs by parts of the wave-packet going around the CI from different sides. 28,31,35

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(s) ˆ e (r; R)hr |φs (R)i = tronic wave-functions |φs (R)i and potential energy surfaces Ee (R): H (s)

Ee (R)hr |φs (R)i. Global adiabatic (GA) representation: The total non-stationary wave-function can be expanded in the adiabatic representation as

hr, R |Ψ(t)i = (s)

where CI

X I,s

(s)

(s)

CI (t)hR |GI i hr |φs (R)i ,

(2)

are time dependent coefficients, indices s and I enumerate the electronic and

nuclear coherent states (CSs) N Y ωj 1/4

h ω j (s) exp − [Rj − qjI (t)]2 π 2 j=1 i (s) (s) i (s) (s) +ipjI [Rj − qjI (t)] + pjI qjI 2

(s)

hR |GI i =

(s)

with time-dependent positions qI

(s)

(s)

= {qjI }j=1,N and momenta pI

(3) (s)

= {pjI }j=1,N [N =

dim(R)]. Equations of motion (EOM) for positions and momenta of CSs can be obtained using TDVP but resulting EOM would introduce unnecessary complexity for our consideration. Thus, here, we adopt simpler EOM that follow classical dynamics on the adiabatic potential energy surfaces (s) (s) q˙ I = pI (s) p˙ I = −

(4)

(s) Ee (R)

∂R

(s)

R=qI

.

(5)

This simplifies variation of the total wave-function by restricting it only to the linear coeffi-

5



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(s)

cients CI (t) δhr, R |Ψ(t)i =

X I,s

(s)

(s)

[δCI (t)]hR |GI i hr |φs (R)i .

(6)

Applying the Dirac-Frenkel TDVP 2,3 ˆ − i∂t |Ψi = 0 hδΨ| H

(7)

and substituting the Ψ and δΨ expressions from Eqs. (2) and (6) we obtain (s′ ) (s′ ) (s) ˆ (ss′ ) (s) − i∂t |GK i CK = 0, δCI hGI | H N

(8)

ˆ e (R) + TˆN |φs′ (R)i ˆ (ss′ ) = hφs (R)| H H N

(9)

X

IK,ss′

where

= Ee(s) (R)δss′ + TˆN + τˆss′ ,

(10)

τˆss′ = − hφs (R)| ∇R φs′ (R)i∇R − hφs (R)| ∇2R φs′ (R)i/2.

(11)

Note that so-called nonadiabatic couplings (NACs) τˆss′ appear as a result of a global dependence of the electronic wave-functions φs on the nuclear coordinates R. Considering (s)

independence of δCI

(s)

variations, EOM for the coefficients CI (t) become X K,s′

(s)

(ss′ )

ˆ hGI | H N

(s′ )

(s′ )

− i∂t |GK i CK = 0,

6


(12)

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Rearranging few terms leads to EOM in the form (s)

iC˙ J

=

X I,K

h (s′ ) (s) ˆ (ss′ ) (s) (s) |GK i hGJ | GI i−1 hGI | H N

i ′ (s ) (s′ ) (s) −i hGI | ∂t GK i CK .

(s)

(13)

(s)

where hGJ | GI i−1 are elements of the inverse CS overlap matrix. Time-derivatives of CSs needed in Eq. (13) are derived using the chain rule (s′ )

|∂t GK

+ + ∂G(s′ ) ∂G(s′ ) ′ K (s ) (s′ ) K i = (s′ ) q˙ K (t) + (s p˙ K (t). ′) ∂q ∂p K K

(14)

The difficulty associated with a proper treatment of the nuclear dynamics using global adiabatic electronic functions is that |φs (R)i are double-valued functions with respect to R in the CI case. To have a single-valued total wave-function in Eq. (2) the nuclear wavefunction must also be double-valued, which is not the case for typical Gaussian-like basis sets [Eq. (3)]. In order to include GP related effects in the nuclear dynamics one needs to substitute the real but double-valued adiabatic electronic wave-functions |φs (R)i by their complex but single-valued counterparts: |φ˜s (R)i = eiθs (R) |φs (R)i, where eiθs (R) is a phase factor that changes its sign when R follows any curve encircling the CI. This phase factor can be seen as a gauge transformation which is needed when a single-valued basis functions for the nuclear counterpart are used. Moving crude adiabatic (MCA) representation: Alternatively, EOM can be derived using a different ansatz for the total wave-function

hr, R |Ψ(t)i =

X I,s

(s)

(s)

(s)

CI (t)hR |GI i hr |φs (qI )i ,

(15)

(s)

here the electronic functions are evaluated only at the centres of CSs, qI , and thus do not (s)

depend on the nuclear coordinates R. To simplify the notation we will denote |φs (qI )i as 7



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(s)

|φI i. Treating CS motion classically [Eqs. (4) and (5)] we repeat the derivation of EOM (s)

for CI (t) in Eq. (15) and obtain (s′′ )

iC˙ J

X

=

I,K

h (s′ ) (s) ˆ (ss′ ) (s) (s) (s′′ ) (s′′ ) hGJ φJ | GI φI i−1 hGI | H IK |GK i

i ′ (s ) (s′ ) (s′ ) (s) (s) −i hφI GI | φK ∂t GK i CK , (s′′ ) (s′′ )

(16)

(s) (s)

where hGJ φJ | GI φI i−1 are elements of the total inverse overlap matrix, and ˆ e (R) |φ(s′ ) i + hφ(s) | φ(s′ ) iTˆN ˆ (ss′ ) = hφ(s) | H H K I K I IK (s)

(17)

(s′ )

−i hφI | ∂t φK i. Here, the adiabatic electronic functions obtained at different points of nuclear geometry and (s)

(s′ )

corresponding to different electronic states are non-orthogonal: hφI | φK i 6= δss′ if I 6= K. Also, the electronic functions are not eigenfunctions of the electronic Hamiltonian for all (s′ )

(s)

ˆ e (R) |φ i is a R and t dependent matrix of functions values of R, therefore, hφI | H K Using the chain rule, the electronic time-derivative couplings in Eq. (17) can be expressed as (s)

(s′ )

hφI | ∂t φK i =

*

+ ∂φ(s′ ) (s) (s′ ) ˙ φI K q ′ K . ∂q(s ) K

(18)

Considering the equivalence between dependencies of the MCA electronic functions on centres of CSs and the GA electronic functions on R, the electronic function derivatives in Eq. (18) are similar to the first order derivative part of NACs in Eq. (11). The first order derivative couplings diverge at the point of the CI, however, since the CS centres form a measure zero subset, CSs will never have their centres exactly at the CI seam. Note that in the MCA representation there are no analogues of the second order derivative parts of NACs. The second order derivatives in NACs pose difficulties for integrating EOM due to their 1/R2 divergent behavior with the distance from the CI R. 36 8


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From the GP point of view, the MCA formalism can be thought as a truly diabatic formalism since the electronic functions do not have the dependence on R, and thus problems emerging in the GA representation do not appear here. Nevertheless, due to a parametric dependence of the adiabatic electronic functions on CSs’ centres, the MCA representation has GPs carried by the electronic functions. We illustrate nuclear dynamics in the introduced representations for a 2D-LVC model where formulated EOM can be simulated without additional approximations and where the GP plays a significant role. The total Hamiltonian for 2D-LVC is 



V12  TˆN + V11 ˆ , HLVC =    V12 TˆN + V22

(19)

where TˆN = − 21 (∂ 2 /∂x2 + ∂ 2 /∂y 2 ) is the nuclear kinetic energy operator, V11 and V22 are the diabatic potentials represented by identical 2D parabolas shifted in the x-direction by a ω2 2 ω2 V22 (R) = 2

V11 (R) =

(x + a)2 + y 2 ,

(x − a)2 + y 2 .

(20) (21)

To have the CI in the adiabatic representation, V11 and V22 are coupled by a linear potential V12 (R) = cy. Thus for this example we have R = (x, y) and the electronic Hamiltonian can ˆ e (R) = P |ϕi i Vij (R) hϕj |, where |ϕi i’s are the diabatic electronic states. be defined as H ij

Switching to the adiabatic representation is done by rotating the electronic basis into the

adiabatic states

|φ1 (R)i =

cos θ(R) |ϕ1 i + sin θ(R) |ϕ2 i ,

(22)

|φ2 (R)i = − sin θ(R) |ϕ1 i + cos θ(R) |ϕ2 i ,

(23)

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which diagonalize the potential matrix. θ(R) is a rotation angle

θ=

1 2 V12 arctan . 2 V22 − V11

(24)

If we track θ changes continuously along a contour encircling the CI, it will change by π, which flips the sign of the phase factor eiθ . 37 The nuclear 2D-LVC Hamiltonian in the adiabatic representation is 





(−)

τˆ12  Ee TˆN + τˆ11 ˆ + Hadi =     0 τˆ21 TˆN + τˆ22



0  ,  (+) Ee

(25)

where Ee(±)

1 1 = (V11 + V22 ) ± 2 2

q (V11 − V22 )2 + 4V122

(26)

are the adiabatic energy surfaces and 1 τˆ11 = τˆ22 = ∇θ · ∇θ 2 1 ∇2 θ + 2∇θ · ∇ τˆ12 = −ˆ τ21 = 2

(27) (28)

are NACs. In order to include the GP we use the gauge transformation of the electronic functions GP ˆ adi eiθ . 30 This that can be seen as a modification of the nuclear Hamiltonian Hadi = e−iθ H

transformation leads to modification of NACs

GP Hadi







(−) E e

GP GP τˆ12 TˆN + τˆ11   + =   GP GP τˆ21 TˆN + τˆ22 0

10




0  ,  (+) Ee

(29)

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where i GP GP τˆ11 = τˆ22 = (∇θ)2 − (∇2 θ + 2∇θ∇), 2 1 GP GP = −ˆ τ21 = −i(∇θ)2 + (∇2 θ + 2∇θ∇). τˆ12 2

(30) (31)

For the MCA representation, the electronic states are calculated as (1)

(1)

(1)

(32)

(2)

(2)

(33)

cos θ(qI ) |ϕ1 i + sin θ(qI ) |ϕ2 i ,

|φI i = (2)

|φI i = − sin θ(qI ) |ϕ1 i + cos θ(qI ) |ϕ2 i , (1)

where qI

(2)

and qI

(s)

(s)

are centres of corresponding CSs. Therefore, integrals hφI | He (R) |φJ i

for the 2D-LVC model are simply linear combinations of Vkl (R) multiplied by cos and sin functions. To illustrate the performance of all three approaches in reproducing the GP we simulate nuclear dynamics of the initial wave-function |φ1 i (1) (1) hR |Ψ(t = 0)i = √ [hR |g1 i + hR |g2 i] 2 (1)

(34)

(1)

that is comprised of two CSs, |GI i = |GI (q, pI )i centred at the same point q = (−1.5, 0) of the ground potential energy surface, but with momenta p1 = (0.1, 0.5) and p2 = (0.1, −0.5), which have the opposite y-components. Using three different Hamiltonians, Eqs. (17), (25), and (29), we simulate time-dependent wave-functions and monitor the total nuclear density ρn (R, t) = Tre [hR |Ψ(t)i hΨ(t)| Ri], where Tre is the trace over the electronic coordinates. Figure 2 illustrates that dynamics with the adiabatic Hamiltonian (25) misses the GP, while two other Hamiltonians reproduce the GP induced destructive interference perfectly. However, mechanisms for the destructive interference in the two approaches is quite different: For the GA representation, two CSs acquire different phases from the −i∇θ∇ part of the diagonal NAC [Eq. (30)] because the θ angle is proportional to the geometric angle between 11




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were very similar to the ones obtained in the current work using the MCA representation. This was the result of approximations needed to make AIMS EOM feasible for simulating dynamics in realistic systems. The current work provides a rigorous framework of the MCA representation that justifies some of the approximations made in AIMS. The MCA representation has been introduced in Ref. 14 and then fully developed in Ref. 15 for the multiconfigurational Ehrenfest approach. Also, the MCA representation can be seen as an effortless realization of a recently proposed on-the-fly diabatization to solve the problem of numerical difficulties in integration of the second order NACs. 41 Acknowledgments:

Authors are grateful to Todd Martinez, Benjamin Levine, and

Michael Schuurman for stimulating discussions, and to the reviewer who brought Ref. 15 to their attention. A.F.I. acknowledges funding from a Sloan Research Fellowship and the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grants Program.

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