Quantum Dynamics through Conical Intersections: Combining

ARTICLE pubs.acs.org/JPCA

Quantum Dynamics through Conical Intersections: Combining Effective Modes and Quadratic Couplings
. Vib
ok,*,† A. Csehi,‡ E. Gindensperger,§ H. K€oppel,|| and G. J. Hal
asz‡ A †

Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, PO Box 5, Hungary Department of Information Technology, University of Debrecen, H-4010 Debrecen, PO Box 12, Hungary § Institut de Chimie, Laboratoire de Chimie Quantique, UMR 7177, CNRS/Universit
e de Strasbourg, 4 rue Blaise Pascal, 67000 Strasbourg, France Theoretische Chemie, Physikalisch-Chemisches Institut, Universit€at Heidelberg, Heidelberg D-69120, Germany

)

‡

bS Supporting Information ABSTRACT: We present a detailed study for the short-time dynamics through conical intersections in molecular systems related to the quadratic vibronic coupling (QVC) Hamiltonian [M€uller, H.; K€oppel, H.; Cederbaum, L. S. New J. Chem. 1993, 17, 7
29] and the effective-mode formalism [Cederbaum, L. S.; Gindensperger, E.; Burghardt, I. Phys. Rev. Lett. 2005, 94, 113003]. Our approach is based on splitting the nuclear degrees of freedom of the whole system into system modes and environment modes. It was found that only three-effective environmental modes together with the system’s modes are needed to describe the short-time dynamics of the complex system correctly. In addition, a detailed mathematical proof is given in the appendix to demonstrate that the exact cumulants are recovered up to the second order within the cumulant expansion of the autocorrelation function. The butatriene molecule is studied as an explicit showcase example to stress the viability of our proposed scheme and to compare with other systems.

1. INTRODUCTION The dynamics taking place in a molecule after absorption of a photon is usually treated in the framework of the Born
Oppenheimer or adiabatic approximation,1,2 which separates the motion of the rapidly moving electrons from the slowly moving nuclei. In this picture, the nuclei move on a single potential energy surface (PES) produced by the faster moving electrons. Although this scheme is suitable to handle several chemical and physical processes, still in many important cases this approximation breaks down. These are nonadiabatic processes where the nuclear and electronic motions can couple, and so-called conical intersections (CIs) play a role.3
8 For these phenomena, the energy exchange between the electrons and nuclei may become significant. CIs between electronic PESs play a key mechanistic role.4
6 In several important processes like dissociation, proton transfer, isomerization of polyatomic molecules, or radiationless deactivation of the excited state systems,9
14 the CIs can provide efficient channels for ultrafast interstate crossing on the femtosecond time scale. The nonadiabatic coupling terms (NACT) couple the different electronic states in the molecule and can be significantly large (as is well-known from the Hellmann
Feynman theorem) in the close vicinity of the CIs.15 They become singular at the CIs thus providing the source for numerous phenomena that are considered as topological effects and lead to several interesting r 2011 American Chemical Society

subjects, including the Longuet
Higgins or Berry phase,16,17 the open-path phase, the quantization feature of the NACTs, etc. CIs can be formed already between low-lying electronic states of triatomic molecules. In truly large polyatomic systems, the CIs are present everywhere. The nonadiabatic dynamics through conical intersections can only be treated correctly in a quantum mechanical framework. The CIs lead to strong nonadiabatic mixing of different vibrational modes on both electronic states. As is well-known from the studies of CIs in polyatomics, the details of the dynamics are highly intricate and cannot be understood without explicit computations. There are plenty of methods in the literature to describe the nuclear dynamics at CIs with high precision, but they can handle correctly only a very limited number of nuclear degrees of freedom.18
23 The most efficient approach to treat the multimode quantum dynamics at CIs is, in our opinion, the multiconfiguration time-dependent Hartree (MCTDH) method.24
26 This scheme is conceptually different from the others as it solves the problem of an approximate nuclear Hamiltonian (linear vibronic coupling Special Issue: Femto10: The Madrid Conference on Femtochemistry Received: July 18, 2011 Revised: October 7, 2011 Published: October 20, 2011 2629

dx.doi.org/10.1021/jp2068528 | J. Phys. Chem. A 2012, 116, 2629–2635

The Journal of Physical Chemistry A (LVC) Hamiltonian3) as accurately as possible. In addition, about 20
30 modes can be described by using it. A few years ago, Cederbaum and co-workers performed a remarkable work, called the three-effective-mode model, to describe the short-time dynamics through CIs in truly large molecular systems.27 This approach concerns only two-state intersections but can be generalized to an arbitrary number of states.31 It both gives a detailed analysis of the different modes of a macromolecule or a molecule embedded in an environment and proposes a scheme of treating them accordingly. This is achieved by an appropriate decomposition of the Hamiltonian. In this approach, all the modes of the macrosystem were decomposed into a system part and an environment part, and then, applying an orthogonal transformation, a new scheme was suggested for further decomposition of the environment modes. Finally, only three effective modes from the environment were obtained, which together with the system modes steer the short time dynamics in macrosystems.28,29 The method obtained permits the performance of accurate quantum dynamical calculations on a short time scale in the close vicinity of conical intersections, describing the effect of the environment with only three effective modes instead of handling all environmental modes explicitly. As a continuation of this work, a further step was suggested by the construction of new additional effective modes, which also permit to describe accurately the intermediate-time dynamics.30
32 By obtaining another conceptually different extension of the presently described effective-mode scheme, we worked out an alternative approach that has been successfully used by Burghardt and collaborators.33
37 The main aim of the present article is to give a further development of the three-effective-mode model. Within this approach, instead of the linear vibronic coupling Hamiltonian (LVC)3 that was used before, the quadratic vibronic coupling Hamiltonian (QVC) will be applied to obtain the appropriate quadratically extended three-effective-mode equations. This QVC Hamiltonian accounts for not only the frequency changes in the interacting electronic manifold but also (implicitly) for a dependence of the vibronic coupling constants on the coordinates of the totally symmetric modes.38 This can give rise to rather complex potential energy surfaces and extend considerably the range of application compared to that of the more traditional LVC scheme (see Chapter seven of ref 5). Indeed, the QVC approach has been applied successfully in the past to reproduce a number of complex photoelectron spectra.39
42 Therefore, it seems very relevant to investigate how it performs in the effective-mode framework. To proceed, the QVC three-effective-mode Hamiltonian is derived and applied to the butatriene molecule as the sample system. The autocorrelation function, the spectrum, and the diabatic populations are calculated and compared to those calculated by the LVC three-effective-mode method. In section 2, the three-effective-mode model is described for the quadratically extended case and characterized. In section 3, we present and analyze the numerical results for the butatriene molecule. In section 4, conclusions are summarized. Finally, in the Supporting Information, a mathematical proof is given for recovering the exact cumulants only up to the second order in the cumulant expansion of the autocorrelation function.

2. METHODOLOGICAL DEVELOPMENT This section starts with the derivation of the working Hamiltonian. It can be obtained by applying the effective-mode

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formalism to the QVC Hamiltonian. As a next step, by using this new Hamiltonian several different properties like autocorrelation function, molecular spectrum etc. will be calculated and discussed. 2.1. Working Hamiltonian. Let us write the N-mode QVC Hamiltonian for a two state conical intersection situation in the diabatic representation ! N E1 0 1 N ðQVCÞ ¼ Hk þ Hk, l ð1Þ H þ 0 E2 2 k, l ¼ 1 k¼1

∑

∑

where

0 ð1, 1Þ kk χk ωk 2 2 @ Hk ¼ ðpk þ χk Þ1 þ ð2, 1Þ 2 kk χk

and

0

Hk, l ¼ @

ð1, 1Þ

ð1, 2Þ

γkl

χk χl

γkl

χk χl

γkl

χk χl

γkl

χk χl

ð2, 1Þ

ð2, 2Þ

ð1, 2Þ

kk

χk

kk

χk

ð2, 2Þ

1 A

ð2Þ

1 A

ð3Þ

Here χk is the coordinate for the kth vibrational mode, pk is the canonical momentum, and 1 is the 2  2 unit matrix. Each individual Hamiltonian Hk is built up of three different parts: The first term is a harmonic 0th-order Hamiltonian with frequency ωk, the second one represents the linear elements which couple the two electronic states, while the third contribution contains the quadratic and bilinear terms. The quantities ki,ik , γi,ik,l and ki,j k, γi,jk,l (i 6¼ j) are the intrastate and interstate coupling constants, respectively. Partitioning the full Hamiltonian in eq 1 into a system Hamiltonian Hsystem and a bath Hamiltonian Hbath, one obtains H ¼ Hsystem þ Hbath where Hsystem ¼ and

E1 0

! þ HS ðy1 , y2 , :::, yNS Þ

0 ð1, 1Þ NB kk χk ωk 2 2 @ ðpk þ χk Þ1 þ ¼ ð2, 1Þ kk χk k¼1 2 k¼1 0 1 ð1, 1Þ ð1, 2Þ 1 NB @ γkl χk χl γkl χk χl A þ 1Þ ð2, 2Þ 2 k, l ¼ 1 γð2, kl χk χl γkl χk χl NB

Hbath

0 E2

ð4Þ

∑

∑

∑

ð5Þ

ð1, 2Þ

kk

χk

kk

χk

ð2, 2Þ

1 A

ð6Þ

This decomposition of the full Hamiltonian can be arbitrary. If the system is a single large polyatomic, the most relevant modes can be collected into the Hsystem Hamiltonian, while the remaining part forms Hbath. However, if our system is a small molecule embedded in an environment, the partition is obvious. 2.2. Effective Modes for the Environment. The construction and the detailed discussion of the effective-mode approach corresponding to the LVC Hamiltonian are given in several papers in the literature.28
31 Next, we consider the Hamiltonian Hbath (eq 6) obtained from the QVC scheme and apply exactly the same type of orthogonal transformation as before. The derivation of the QVC effective-mode Hamiltonian will be quite similar to the former (LVC) one. 2630

dx.doi.org/10.1021/jp2068528 |J. Phys. Chem. A 2012, 116, 2629–2635

The Journal of Physical Chemistry A

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Let us decompose the operator Hbath into two parts. For doing this, it is useful to introduce a unitary transformation of the bath modes that splits Hbath into components Heff and Vbath as follows: Hbath ¼ Heff þ Vbath

At this stage, we are able to derive the effective Hamiltonian envisaged in eq 7. For the sake of completeness we also present here the form of the operator Vbath . These operators are

ð7Þ

Heff ¼ εeff þ HA

þ HB

In this separation, the sum of the Hsystem and Heff Hamiltonians,

Vbath ¼ εbath þ Hb

þ Hcði, jÞ þ Hd

H 0 ¼ Hsystem þ Heff

ði, jÞ

ði, jÞ

ði, jÞ

ði, jÞ

ði, jÞ

ði, jÞ

ði, jÞ

ði, jÞ

ði, jÞ

þ HC

ði, jÞ

þ HD

ði, jÞ

þ HE

þ Heði, jÞ

ði, j ¼ 1, 2Þ

ð8Þ

ð14Þ

steers the short-time dynamics of the system, while the remaining part, Vbath, has to be taken into account only at later times. It is noticed that the Heff operator consists only from three (effective) modes, which couple the two electronic states. Let us define (1,1) (1,2) (2,1) B B B ~ 1, ∑N χk = k̅ (1,1) X χk = ∑N the elements ΣN k=1 kk k=1 kk k=1 kk (1,2) ~ (2,2) ~ NB (2,2) χk = k̅ X 2, and ∑k=1 kk χk = k̅ X 3 which appear in Hbath (1,1) 2 1/2 B as effective modes. The expressions k̅ (1,1)  (∑N )) , k=1 (kk (1,2) (2,2) NB (1,2) 2 1/2 NB (2,2) 2 1/2  (∑k=1 (kk ) ) , and k̅  (∑k=1 (kk ) ) denote k̅ the effective coupling constants.28
31 First, they are not orthogonal to each other, and second, they do not have any physical importance either. Nevertheless, these terms can be written as the linear combination of three orthogonal modes. The construction of them is as follows:28
31 ~ 1, X ~ 2, X ~ 3 ÞT ðX1 , X2 , X3 ÞT ¼ U33 ðX ¼ U33 V3NB ðχ1 , χ2 , :::, χNB ÞT ð9Þ

The appropriate forms for the different terms of the oper(i,j) ators H(i,j) eff and Vbath are collected below

¼ T 3NB ðχ1 , χ2 , :::, χNB ÞT Here, the X1, X2, and X3 vectors are normalized and orthogonal to each other, and U33 is a matrix that orthogonalizes the ~ l, l = 1,2,3. Combining this matrix U33 with the modes X matrix V3NB, one can obtain the T3NB transformation matrix (U33
1 = V3NB T3  NBT) between the initial environmental modes and corresponding orthonormalized ones. The V3NB transformation matrix gives the connection between the initial environmental modes and the intermediate normalized ones ~ 2, X ~ 3)T = V3N (χ1,χ2, ...,χN )T, where ~ 1, X (X B B 0 1 ð1, 1Þ ð1, 1Þ ð1, 1Þ ð1, 1Þ k1 =k ̅ ::: kNB =k ̅ B C B ð1, 2Þ ð1, 2Þ ð1, 2Þ C ð1, 2Þ C ̅ ::: kNB =k ̅ V3NB ¼ B k1 =k ð10Þ @ A ð2, 2Þ ð2, 2Þ ð2, 2Þ ð2, 2Þ k1 =k ̅ ::: kNB =k ̅ By applying the transformation T 3N B to the original (i,j) B χk and modes of the environment, the expressions ∑N k=1 kk NB (i,j) (1/2) ∑k,l=1 γkl χkχl of Hbath(eq 6) can be written as NB

3

ði, jÞ ði, jÞ ði, jÞ kk χk ¼ k̅ Kk X k ∑ ∑ k¼1 k¼1

1 NB ði, jÞ γ χχ ¼ 2 k, l ¼ 1 kl k l

∑

NB

∑ k¼1

ði, jÞ

dkk 2 X þ 2 k

ð11Þ NB

ði, jÞ dkl Xk Xl ∑ k, l ¼ 1

ð12Þ

k