Car-Parrinello Molecular Dynamics on excited state surfaces

Abstract

This paper describes a method to do ab initio molecular dynamics in electronically excited systems within the random phase approximation (RPA). Using a dynamical variational treatment of the RPA frequency, which corresponds to the electronic excitation energy of the system, we derive coupled equations of motion for the RPA amplitudes, the single particle orbitals, and the nuclear coordinates. These equations scale linearly with basis size and can be implemented with only a single holonomic constraint. Test calculations on a model two level system give exact agreement with analytical results. Furthermore, we examined the computational efficiency of the method by modeling the excited state dynamics of a one-dimensional polyene lattice. Our results indicate that the present method offers a considerable decrease in computational effort over a straight-forward configuration interaction (singles) plus gradient calculation performed at each nuclear configuration.

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arXiv:cond-mat/9805271v3 22 Jan 1999
Car-Parrinello molecular dynamics on excited state surfaces
Eric R. Bittner and D.S. Kosov∗
Department of Chemistry
University of Houston
Houston, TX 77204, USA
(February 1, 2008)
This paper describes a method to do ab initio molecular dynamics in electronically excited
systems within the random phase approximation (RPA). Using a dynamical variational treatment
of the RPA frequency, which corresponds to the electronic excitation energy of the system, we derive
coupled equations of motion for the RPA amplitudes, the single particle orbitals, and the nuclear
coordinates. These equations scale linearly with basis size and can be implemented with only a
single holonomic constraint. Test calculations on a model two level system give exact agreement
with analytical results. Furthermore, we examined the computational efficiency of the method by
modeling the excited state dynamics of a one-dimensional polyene lattice. Our results indicate that
the present method offers a considerable decrease in computational effort over a straight-forward
configuration interaction (singles) plus gradient calculation performed at each nuclear configuration.
Pacs: 71.15-m, 71.15.Pd, 31.50.+w
I. INTRODUCTION
In recent years, there has been considerable advances towards fully ab initio molecular dynamics simulations using
inter- and intramolecular potentials derived from quantum many-body interactions. This activity has been largely
spurred by the introduction of hybrid quantum/classical dynamics treatments, such as the density functional the-
ory/molecular dynamics methods introduced by Car and Parrinello1–3(CPMD). In this approach, the quantum elec-
tronic orbitals, {φi(1)}, are coupled to the classical nuclear degrees of freedom, Rn, through a fictitious Lagrangian,
which along with an appropriate set of Lagrange multipliers, (Λij), leads to a set of equations of motion
µ¨φi(1) = −δE[{φi},Rn]
δφ∗
i(1)
+
?
k
Λikφk(1),
Mn¨Rn= −∇RE[{φi},Rn],(1)
where E[{φi},Rn] is the Hohenberg and Kohn energy functional4. This approach has proven to be highly successful
in simulating chemical dynamics in the condensed phase.
However, there are a number of distinct disadvantages to the method. First, the equations of motion involve both
fast (the orbitals) and slow (the nuclei) degrees of freedom. Thus, the computational time-step must be rather small.
Secondly, imposing the holonomic constraint of orbital orthogonality forces a single iteration of the approach to scale
as ∼ M ×N2where N is the number of particles and M the size of the electronic basis. Finally, the method can only
treat systems in the electronic ground state.
Several groups have attempted to overcome each of these difficulties. The time-step problem has been attacked
via multiple time-scale methods5,6and by using the electronic density instead of the Kohn-Sham7orbitals as the
dynamical variable8. Several algorithms have been introduced to tackle the scaling problem3. In these cases, scaling
more favorable than ∼ N3is achieved only at the cost of several assumptions and approximations.
Recently, Alavi, Kohanoff, Parrinello, and Frenkel have developed a density functional based method for treating
systems with finite temperature electrons9,10. This approach has a number of attractive features, foremost being
the inclusion of fractional occupation of the electronic orbitals. Dynamics in this scheme are isothermal rather than
adiabatic, allowing for incoherent transitions between electronically excited states. While the approach works well for
small bandgap systems (such as metals or dense hydrogen), this approach is not useful for systems with larger band
gaps (∆E ≫ kT) or photo-excited systems.
In this paper, we present an alternative formulation of the Car-Parrinello method for treating electronically excited
systems. We begin by introducing the general computational algorithm for the dynamical optimization of excited
electronic states in the presence of “classically propagating” ions. This general theoretical scheme is developed without
∗Present address: Department of Chemistry, UMIST, Manchester, M60 1QD, UK
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specific reference to the description of the electronic excited state. The principal results of this section are a set of
dynamical equations of motion for the electronic amplitudes and nuclear coordinates in a generalized phase space.
We then derive specific equations of motion for the electronic amplitudes and nuclear coordinates under the Random
Phase Approximation (RPA) for the electrons. Finally, we demonstrate the salient features of the approach through
a simplified SU(2) model of interacting electrons and via a realistic model of conjugated one-dimensional polymer
lattices.
II. CLASSICAL EQUATIONS OF MOTION FOR THE DYNAMICAL OPTIMIZATION OF EXCITED
QUANTUM STATES
We begin by developing a general dynamical theory for molecular dynamics in excited electronic states. We first
define the N-electron Fock space as being parameterized by a set of dynamical variables λ(τ) which include the
nuclear positions and any other dynamical variables and ensemble constraints placed upon the system (e.g. pressure,
temperature, volume etc...):
F =
?
|n1,n2,...(λ)?,
?
i
ni= N
?
(2)
We describe the ground state wave function as an expansion on the basis in F:
|Ψo(λ)? =
?
1...i...
Ci(λ)|n1...ni...(λ)? ,?Ψo(λ)|Ψo(λ)? = 1(3)
|Ψo(λ)? is not a Slater determinant in general case.
We next define the mapping operators, P†
orthogonal to |Ψo(λ)? in the subspace F+of the Fock space:
ν, which act on the ground state to produce a new state, |ν(λ)?, which is
|ν(λ)? = P†
ν|Ψo(λ)?, (4)
Pν|Ψo(λ)? = 0, (5)
?ν(λ)|Ψo(λ)? = ?Ψo(λ)|Pν|Ψo(λ)? = 0 ,?ν(λ)|ν(λ)? = 1(6)
If the |Ψo(λ)? is a true ground state of the N-particle system, the excited states should lie in the orthogonal subspace
F+11.
Variational determinations of the excited states are thus searches in the orthogonal subspace for the optimal mapping
operator P†
νand for the optimal basis of the Fock space |n1n2...ni..? such that the energy functional
εν= ?ν(λ)|H|ν(λ)?,
= ?Ψo(λ)|H|Ψo(λ)?
+ (?ν(λ)|H|ν(λ)? − ?Ψo(λ)|H|Ψo(λ)?),
= ?Ψo(λ)|H|Ψo(λ)? + ?Ψo(λ)|?Pν,?H,P†
= Eo+ ων,
ν
??|Ψo(λ)?,
(7)
is at a variational minimum. We write the excitation energy, εν, as a functional of both the ground and excited state
and introduce explicitly via the double commutator term the energy gap, ων, between the ground and the excited
state.
According to the calculus of variations, we can derive equations of motion which dynamically optimize the excited
state system. We first define formally a set of “velocities” of the states in the Fock space
|˙ ν? =
d
dτ|ν? , |˙Ψo? =
d
dτ|Ψo?,(8)
where τ is not the true physical time but rather a parameter which characterizes the evolution of the amplitudes
through the Hilbert space subject to the orthogonalization constraints given above. The “classical” Lagrangian for
our system is given by
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L =1
− εν[|0?,|ν?,λ]
− Λ(?Ψo|Ψo? − 1) − Γ(?ν|ν? − 1),
2µ|˙Ψo??˙Ψo| +1
2µ′|˙ ν??˙ ν| +1
2m˙λ2(τ)
(9)
where Λ and Γ are Lagrange multipliers arising from the holonomic ortho-normalization constraint for the ground and
excited states and which allow |ν? and |Ψo? to be treated as independent variables. The masses µ and µ′are fictitious
masses and have no true physical meaning. This Lagrangian leads to the classical Euler-Lagrange equations12for the
wave functions:
d
dτ
∂L
∂|˙Ψo?−
d
dτ
∂L
∂|Ψo?= 0,
∂L
∂|ν?= 0.
(10)
∂L
∂|˙ ν?−
(11)
The Euler-Lagrange equation dictates the following equations of motion for quantum dynamical optimization in the
Hilbert space:
µ|¨Ψo? = −δεν
µ′|¨ ν? = −δεν
m¨λ = −δεν
δ?Ψo|+ Λ|Ψo?,
δ?ν|+ Γ|ν?,
δλ.
(12)
This dynamical view is the starting point for the remainder of our calculations.
III. RANDOM PHASE APPROXIMATION FOR ELECTRONIC MOTIONS
For a molecular system with N electrons and n nuclei, the full N-body Hamiltonian for a fixed set of nuclear
positions {Rν}, is given as (atomic units are used throughout)
H =1
2
i
ν
i
?
N
?
∇i2−
??
1
Zν
|Rν− ri|+
?
ij
1
|ri− ri|, (13)
=
i
h(i)
◦ +
?
ij
|ri− ri|,(14)
where h(i)
the electron and the nuclear cores. We define the single-particle Hamiltonian, H1, through either the Hartree-Fock
(HF) approximation13or via the density functional theory of Kohn and Sham7. In either case, H1is a functional of
the single particle density
◦
is the one-body electron operator which includes the electron kinetic energy and the interaction between
ρ(12) =
occ
?
i
φ∗
i(1)φi(2).(15)
The single particle states, {φi(1)}, are solutions of a nonlinear Schr¨ odinger equation,
H1[ρ,R]|φi? = Ei[R]|φi?,(16)
which must be determined at the start of the calculation. Here Ei[R] is the orbital energy. The HF spin-orbital wave
functions and energies differ from the KS ones in the way the terms in the single-particle Hamiltonian are treated.
We will denote the Slater determinant wave-function constructed from occupied single-particle HF (of KS) orbitals
as |HF? and note that this state represents either the Hartree-Fock ground state or the Kohn-Sham ground state.
Having obtained a basis of single-particle states, we can write the two-body Hamiltonian in second quantized form:
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H = EHF+
?
(ij|v|kl) : a†
iσ
Ei: a†
iσaiσ:
+1
2
?
ijkl
?
σσ′
iσa†
jσ′alσakσ′ :,(17)
where Eiis the HF single-particle eigenenergy,
(ij|v|kl) =
?
d1d2φ∗
i(1)φ∗
j(2)v(12)φk(1)φl(2),(18)
is the matrix element of Coulomb electron-electron interactions, : ··· : denotes a normal ordered product and EHF is
the HF ground state energy. The operators a†
iσ(aiσ) create (annihilate) single electrons in the {iσ} HF spin-orbital
state, where σ = ±1
It is convenient at this point to move to a particle/hole representation and introduce the particle/hole quasi-particle
operator,
2is the spin index of the electrons.
αiσ|HF? = 0,(19)
through its action on the HF vacuum, where i = p,h represents particles (occupied states above the Fermi energy)
and holes (vacancies below the Fermi energy). These quasi-particles operators are related to the fermion operators
written above via
a†
iσ= θ(Ei− Ef)α†
iσ+ θ(Ef− Ei)αiσ,(20)
aiσ= θ(Ei− Ef)αiσ+ θ(Ef− Ei)α†
iσ, (21)
where θ(x) is the Heaviside step function and Ef is the Fermi energy.
We can expand H in terms of the particle/hole operators and neglect terms with an odd number of particle or hole
terms (i.e. anharmonic interaction terms of the form ∼ α†ααα and ∼ α†α†α†α),
?
+1
2
p1p2h3h4
?
+
?
−
p1h2p3h4
+ anharmonic terms,
H = EHF+
pσ
Epα†
pσαpσ+
?
hσ
Ehαhσα†
hσ
?
α†
(p1p2|v|h3h4)
×
?
σσ′
p1σα†
p2σ′α†
h3σα†
h4σ′ + h.c.
?
h3σαh2σ′αp4σ′
p1h2h3p4
(p1h2|v|h3p4)
?
?
σσ′
α†
p1σα†
?
(p1h2|v|p3h4)
σσ′
α†
p1σα†
h4σ′αh2σ′αp3σ
(22)
where p and h refer to particle and hole single particle states.
Note that each “harmonic” term in our expansion of H involves an even number of particle and hole operators.
We can form a new set of exciton operators by taking bilinear combinations of the particle/hole operators with total
angular momentum S and projection mS,
A†
ph(SmS) =
?
?
σ1σ2
?1
2σ11
2σ2|Sms?α†
pσ1α†
hσ2,
Aph(SmS) =
σ1σ2
?1
2σ11
2σ2|SmS?αhσ2αhσ1,(23)
where ?s1σ1s2σ2|Sms? is a Clebsch-Gordon coefficient. Our notation is such that σ denotes the action of time reversal
operator on the hole state,
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α†
hσ= (−)
1
2+σα†
h−σ. (24)
The {A†
Writing just the “harmonic” part of H in terms of exciton operators,
ph(SmS),Aph(SmS)} serve to create (or destroy) particle/hole exciton with spin S and projection ms.
H2=
?
pσ
Epα†
pσαpσ+
?
hσ
Ehαhσα†
hσ
−1
2
?
?
p1p2h3h4
(p1p2|v|h3h4)
?
?
Sms
(−)S?
?
A†
p1h4(Sms)Ap3h2(Sms),
A†
p1h4(Sms)A†
p2h3(Sms) + h.c.
?
+1
2
p1h2h3p4
(p1h2|v|h3p4)
Sms
S′m′
s
?(−)S+ 1??
(−)S
′
+ 1
?
A†
p1h3(Sms)Ap4h2(S
′m
′
s)
−
?
p1h2p3h4
(p1h2|v|p3h4)
?
Sms
(25)
where Aph(Sms) = (−)S+msAph(S − m) is the time reversal exciton annihilation operator. This defines our “quasi-
harmonic” exciton Hamiltonian.
The exact commutation relation between the exciton operators is
[ Aph(Sµ),A†
p′h′(S′µ′)]
?1
=
?
σ1σ2
?
σ1′σ2′
2σ11
2σ2|λµ??1
2σ′
1
1
2σ′
2|λ′µ′?
× (αhσ2α†
h′σ′
2δpp′δσ1σ1′ − α†
p′σ′
1αpσ2δhh′δσ2σ2′).(26)
We apply the Wick theorem to the right hand side of Eq. 26 and then neglect the normal ordering contribution to
this relation. This is the RPA14and in this approximation the commutation relation between the exciton operators
is bosonic
[Aph(Sµ),A†
p′h′(S′µ′)] ≈ δSS′δµµ′δhh′δpp′. (27)
We then perform a canonical Bogolubov transformation15for both singlet (S = 0) and triplet (S = 1) exciton bosons
Q†
i(00) =
?
?
ph
Xi
phA†
ph(00) − Yi
phAph(00),
Q†
i(1ms) =
ph
Xi
phA†
ph(1ms) − Yi
phAph(1ms), (28)
where Xi
harmonic excitations in the electronic degrees of freedom. Physically, the singlet RPA excitations correspond to
“breathing” modes of the electron cloud and triple RPA excitations correspond to dipolar oscillations of the electronic
system about the nuclear frame. In what follows, we will focus our attention upon the triplet excitations, as these are
the states created upon photo-excitation of a singlet ground state. Equations of motion for singlet RPA excitations
can be obtained as well.
The commutation relations for the RPA excitation creation operators goes as follows:
phand Yi
phare the RPA amplitudes which are yet to be determined. These operators create and destroy
[Qµi,Q†
µ′i′] =
?
ph
Xi
ph
∗Xi′
ph− Yi
ph
∗Yi′
ph
= δii′δµµ′. (29)
By defining the RPA vacuum as
Qµi|RPA? = 0, (30)
and the RPA excited state as
Q†
µi|RPA? = |ωµi?, (31)
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