A Hamiltonian-Independent Generalization of the Fragment Excitation

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A Hamiltonian-Independent Generalization of the Fragment Excitation Difference Scheme Karl Y Kue, Gil C Claudio, and Chao-Ping Hsu J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01103 • Publication Date (Web): 22 Jan 2018 Downloaded from http://pubs.acs.org on January 28, 2018

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Journal of Chemical Theory and Computation

A Hamiltonian-Independent Generalization of the Fragment Excitation Difference Scheme Karl Y. Kue,†,‡ Gil C. Claudio,† and Chao-Ping Hsu∗,‡ †Institute of Chemistry, University of the Philippines Diliman, Quezon City, 1101, Philippines ‡Institute of Chemistry, Academia Sinica, 128 Section 2 Academia Road, Nankang, Taipei, 115, Taiwan E-mail: [email protected] Abstract The fragment excitation difference (FED) scheme is a useful method for calculating the complete diabatic couplings of various energy transfer systems. The lack of a good definition for the transformation of the transition density matrix to the off-diagonal FED matrix elements limits FED to single-excitation methods. We have developed a generalized FED scheme called θ-optimized FED (θ-FED) scheme which does not require transforming the transition density matrices. In θ-FED, two states of interest are linear transformed by a mixing angle θ into two mixed states. The excitation difference of each mixed state is evaluated and optimized numerically to determine the mixing angle. This approach allows for finding diabatic states and the corresponding couplings for a general set of Hamiltonians.

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Introduction

When a molecule or fragment is electronically excited, the excitation may be transferred to a nearby molecule or fragment, a process called excitation energy transfer (EET). Molecular EET is important in both natural and artificial light-harvesting systems. EET is the fundamental process underlying light-harvesting in photosynthesis, specifically, the very first event upon the absorption of light. 1,2 Similar concepts have also been used in the design of solar cells and light-emitting devices. 3–5 Proper characterization of EET is important for deriving insights into the observed processes and for designing better organic electronics. 5–8 The rate of EET can be modeled by using Fermi’s golden rule

k=

2π 2 |V | (FCWD) h ¯

(1)

where FCWD is the Frank-Condon weighted density of states and V is the electronic coupling. The electronic coupling is normally the off-diagonal Hamiltonian matrix element of the diabatic states V = hΨi |H|Ψf i

(2)

where Ψi and Ψf are the diabatic wave functions for the initial and final states. The diabatic states originate from the break-down of the Born-Oppenheimer approximation, where the two energetically close states (as is involved in EET processes) are coupled through the motion of the nuclei. In describing the dynamics, diabatic states are states with zero offdiagonal elements for the derivative of the nuclear degrees of freedom. For systems larger than diatomics, the zero derivative coupling condition is over-determined. 9 In other words, strictly diabatic states generally do not exist. Approximated diabatic states can be alternatively defined where certain off-diagonal derivative couplings are zero or close to zero. 10 There are other ways to define and calculate the diabatic states, hence the couplings. Diabatic states can be constructed by requiring a “smooth” evolution of a chosen physical property along the potential surface. 11,12 The smooth requirement is achieved by diagonal2


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ization of the matrix of the corresponding property operators, which also leads to the optimal separation of that property-measurement in the diabatic states. 12 Therefore, it is common to form a linear combination of adiabatic wave functions that fulfill a localization condition or by imposing a separation in the calculation. Examples of the former include Boy’s localization 13 and Edmiston-Ruedenberg diabatization, 14 and examples of the latter include block localization 15 and the constrained density functional theory, 16 among other similar approaches such those benchmarked in ref. 17. A general way to formulate diabatic states is to use an orthonormal linear combination of two adiabatic states,

Ψi = Ψ1 cos θ + Ψ2 sin θ

(3)

Ψf = −Ψ1 sin θ + Ψ2 cos θ

(4)

where Ψ1 and Ψ2 are adiabatic states and θ is a mixing angle; this method requires only a single constraint. One such single constraint method is the fragment excitation difference (FED) scheme for calculating EET coupling. 18 FED is a generalization of the generalized Mulliken-Hush (GMH) 12,19,20 and fragment charge difference (FCD) 21 schemes for electron transfer problems. 8,22,23 In these schemes, the coupling is calculated by using the localized diabatic states with the largest excitation (FED), dipole (GMH), or charge (FCD) difference. In FED, the diabatic states are formed by first calculating the excitation difference matrix 



 ∆x1 ∆x12  ∆x =   ∆x12 ∆x2

(5)

where ∆x1 , ∆x2 are the excitation differences of the first and second adiabtic states of interest, defined as Z ∆xm =

ρ(m) ex (r) dr

r∈D

Z −

ρ(m) ex (r) dr

(6)

r∈A

(m)

where ρex (r) is the excitation density which is defined as the sum of the electron and hole 3



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densities. ∆x12 is the corresponding quantity evaluated for the transition density from state 1 to state 2. Hsu et al. 18 proposed that the excitation density can be defined as the sum of electron and hole densities calculated from the attachment and detachment densities 24 respectively. The diabatic states are those that diagonalize the matrix ∆x ∆x, 



∆xi 0  U −1 ∆xU =   0 ∆xf

(7)

where the rotation matrix U can be written as   cos θ − sin θ U =  sin θ cos θ

(8)

and θ in eq. (8) is the same θ in eqs. (3) and (4). The coupling is then obtained by applying the rotation matrix to the energy matrix 







E i V   E1 0  U −1   U =  V Ef 0 E2

(9)

where E1 , E2 are the energies of the adiabatic states, Ei , Ef are the energies of the diabatic states, and V is the coupling in eq. (1). The EET coupling can be approximately separated into a Coulomb and an exchangeoverlap term, 18,25–27 with the former being mostly dominant. When the diabatic states are modeled by a combined ground- and excited-state donor and acceptor without considering their interaction, a direct Coulomb coupling of the transition densities can be a good approximation for EET coupling. 28–30 This approximation has a long history in theoretical development. The F¨orster theory is based on the leading moment expansion for the Coulomb coupling, i.e. the dipole-dipole coupling. 31 The fragment spin difference (FSD) 32 is a similar scheme developed for predicting the

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rates of triplet EET (TEET), where ∆x in eq. (5) is replaced with the spin density difference derived from the differences in the α and β electron densities of the fragments. Such a scheme has been used to successfully model the TEET in light-harvesting complexes. 32–34 Recently, Yang and Hsu 35 successfully used the FSD scheme along with the spin-flip approach 36 to calculate singlet fission couplings. Hence, the fragment difference schemes have applications beyond excitation transfer. The FED scheme relies on calculating the electron and hole densities of the adiabatic states and of the transition densities. These densities are calculated by using attachment and detachment densities. 24 The original formulation of the attachment and detachment densities applied to only state densities and cannot be easily generalized for transition densities when multiple-electron excitations are included. We propose an alternate scheme called the θoptimized-FED (θ-FED) scheme to calculate FED diabatic states and FED couplings without the need to calculate the excitation density, thus the attachment and detachment densities, of the transition density. We show that this method is equivalent to FED when applied to single-excitation wave functions and can be used for multi-excitation wave functions.

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Theory

The main assumption for FED is that the diabatic states are the orthonormal linear combinations of two adiabatic states with the most fragment-localized excitation. We developed the θ-FED scheme to achieve the same localization without having to calculate the off-diagonal fragment excitation components.

2.1

FED with single-excitation theories

x to form the diabatic states The FED scheme relies on the excitation difference matrix ∆x (eq. (5)). In the original work of Hsu et al., 18 the electron and hole densities are described by using the attachment and detachment densities. 24 FED was originally implemented for the

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single-excitation wave functions such as configuration interaction (CIS) wave functions 18 and recently for spin-flip CIS (SF-CIS) wave functions. 35 Both single-excitation wave functions can be written as Ψ=

X

aai Φai

(10)

ia

where the index i refers to an occupied orbital and the index a refers to a virtual orbital. The methods only differ in the choice of index i and a. In these wave functions, the need for an eigendecomposition of the difference densities to obtain the attachment and detachment densities can be bypassed because the difference densities are composed of a positive definite block and a negative definite block. The attachment density can be taken as the positive definite block, calculated as (m) ρelec (r)

=

occ vir X X ab

(m) (m)∗

aia aib

ψa (r)ψb∗ (r)

(11)

i

and the detachment density can be taken as the negative definite block, calculated as (m) ρhole (r)

=

vir occ X X ij

(m) (m)∗

aia aja ψi (r)ψj∗ (r)

(12)

a

For the transition density, the electron and hole densities can be obtained by applying the same equations (eqs. (11) and (12)) and replacing a(m)∗ with the amplitude of the second excited state, for example a(n)∗ , (mn) ρelec (r)

=

vir X occ X ab

(m) (n)∗

(13)

(m) (n)∗

(14)

aia aib ψa (r)ψb∗ (r)

i

and, (mn) ρhole (r)

=

occ X vir X ij

aia aja ψi (r)ψj∗ (r)

a

In this case, the contributions of the amplitudes—whether to the hole density or to the electron density—are clear.

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For multi-excitation wave functions, no expressions such as eqs. (11) to (14) exist and the attachment and detachment densities are obtained via eigendecomposition. 24 For the diagonal difference excitation elements, extracting the positive definite and negative definite parts of the difference densities is straightforward because they are Hermitian. However, a corresponding treatment for the transition densities does not exist. The same eigendecomposition for the transition densities would yield complex eigenvalues because they are generally non-Hermitian matrices. In the present work, we propose an alternate method to obtain the excitation-localized state without the need to calculate the electron and hole densities of the transition density.

2.2

θ-FED

Instead of attempting to characterize the excitation transition density and diagonalizing the x matrix to obtain the diabatic states, we let the diabatic states be functions of the mixing ∆x angle θ as defined in eqs. (3) and (4). Consequently, the diabatic difference, attachment, detachment, and excitation densities are dependent on θ and can be written as Z ∆xii (θ) =

ρ(ii) ex (r, θ) dr

r∈D

Z −

ρ(ii) ex (r, θ) dr

(15)

r∈A

where ii = i or f. The terms in eq. (15) can be obtained similar to the adiabatic equivalent because only the coefficients of the wave functions are changed; the diabatic difference densities will still be Hermitian because they are sums of two Hermitian matrices. We note that although the diabatic excitation population is written as a function of θ, a closed-form expression cannot be obtained because its value depends on an eigendecomposition of the diabatic difference densities. We propose that the ideal diabatic states will be the ones with a maximum excitation difference. That is, the value of θ is chosen such that the difference of the excitation population

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of the two diabatic states is maximized,

θmax = arg max ∆xi (θ) − ∆xf (θ)

(16)

−π/4

A Hamiltonian-Independent Generalization of the Fragment Excitation

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