Superoperator many-body theory of molecular currents: non ...mukamel.ps.uci.edu/publications/pdfs/509.pdf · ex ¼ ð drjðr;tÞc†ðrÞcðrÞð6Þ We next brieﬂy survey some properties

CHAPTER 14

Superoperator many-body theory of

molecular currents: non-equilibrium

Green functions in real time

Upendra Harbola and Shaul Mukamel

Department of Chemistry, University of California, Irvine, CA 92697-2025, USA

Abstract

The electric conductance of a molecular junction is calculated by recasting the Keldysh

formalism in Liouville space. Dyson equations for non-equilibrium many-body Green

functions (NEGF) are derived directly in real (physical) time. The various NEGFs appear

naturally in the theory as time-ordered products of superoperators, while the Keldysh

forward/backward time loop is avoided.

14.1 INTRODUCTION

Recent advances in the fabrication and measurements of nanoscale devices have led to a

considerable interest in non-equilibrium current carrying states of single molecules. The

tunneling of electrons between two metals separated by a thin oxide layer was first

observed experimentally by Giaever [1] and later by others [2]. Vibrational resonances

can be observed for molecules absorbed at the metal–oxide interface by analyzing the

tunneling current as a function of the applied bias [3,4]. More recent development of

scanning tunneling microscopy (STM) led to a direct, real space determination of surface

structures. A metal tip is brought near the surface so that tunneling resistance is

measurable. A contour map of the surface is obtained by recording the tunneling

resistance as the tip scans the surface. The tunneling electrons interact and may exchange

energy with the nuclear degrees of freedom of the absorbed molecule. This opens up

inelastic channels for electron transmission from the tip to the surface, leading to inelastic

electron tunneling (IET). IET may play an important role in manipulating molecules with

STM [5,6]. Recently, IET was combined with STM for the chemical analysis of a single

absorbed molecule with atomic spatial resolution [7,8]. The atomic scale images of STM

q 2005 Elsevier B.V. All rights reserved.

Theory and Applications of Computational Chemistry

Edited by C. Dykstra et al. 373

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References pp. 395–396

have been useful in analyzing different binding configurations of the molecule

chemisorbed on a semiconductor surface [9,10]. Recent advances in the theory of

STM are reviewed in Ref. [11].

Electron tunneling was first analyzed by Bardeen [12] and Cohen et al. [13] using the

perturbative transfer Hamiltonian (TH) approach and more recently by many other

authors [14–16]. Although the TH gives, in most cases, a good description of the

observed effects, it lacks a firm first principles theoretical basis and does not account

properly for many-body effects [17]. An improved form of TH [18] that involved energy

dependent transfer matrix elements was used to incorporate many-body effects. However,

this model does not describe the electron–phonon interaction properly [19].

A many-body non-equilibrium Green functions (NEGF) formulation of electron

tunneling was proposed by Caroli et al. [20]. The NEGF theory was originated by

Schwinger [21] and Kaddanof and Baym [22], and developed further by Keldysh [23] and

Craig [24]. This formalism involves the calculation of four basic Green functions, time

ordered ðGT Þ; anti-time ordered ðG~TÞ; lesser ðG,Þ and greater ðG.Þ: Additional retarded

(Gr) and advanced (Ga) Green functions are defined as specific combinations of these

basic functions. At equilibrium suffice it to know only the retarded or advanced Green

functions; all other Green functions simply follow from the fluctuation–dissipation

theorem that connects the ‘lesser’ and ‘greater’ with the retarded Green function through

the equilibrium Fermi distribution function ðf0ðEÞÞ [25]. However, for non-equilibrium

Q1

measurements, where the distribution function is not known a priori, one needs to solve

for the various NEGFs self-consistently.

Electronic transport in molecular wires and STM currents of single molecules have

received considerable attention [26–31]. Electron transport through a single molecule

[32–34] or a chain of several atoms [35] was studied. From a theoretical point of view,

this is very similar to the electron tunneling in semiconductor junctions and various

theories developed for STM [36,37] can directly be applied to molecular wires. The

NEGF technique developed for tunneling currents has been used to analyze the electron

conduction through a single molecule attached to electrodes [26,36,38–42]. The method

has also been combined with density functional theory for the modeling of transport in

molecular devices [43,44].

In this chapter, we develop a non-equilibrium superoperator Green function theory

[23,25,45] (NESGF) of molecular currents [46]. A notable advantage of working with

superoperators in the higher dimensional Liouville space [44,47] is that we need to

consider only time-ordered quantities in real (physical) time; all NEGFs show up

naturally without introducing artificial time variables. Observables can be expressed in

terms of various Liouville space pathways (LSP) [46]. The ordinary (causal) response

function which represents the density response to an external field is one particular

combination of these LSPs. Other combinations represent the spontaneous density

fluctuations and the response of these fluctuations to the external field [47,48]. A simple

time ordering operation of superoperators in real time is all it takes to derive the non-

equilibrium theory, avoiding the Keldysh loop or Matsubara imaginary time. The NESGF

theory provides new physical insights into the mechanism of the current. It can also be

more naturally used to interpret time domain experiments involving external pulses.

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In Section 14.2, we give a brief introduction to the superoperator formalism and recast

the NEGF theory in terms of the superoperator Green functions. Starting from the

microscopic definitions for various NESGF, we construct dynamical equations of motion

and obtain the Dyson matrix equation of Keldysh which couples the various NESGFs. In

Section 14.3, we apply the NESGF theory to the conduction through a molecular

junction. In Section 14.4 we end with a discussion.

14.2 DYSON EQUATIONS FOR SUPEROPERATOR GREEN FUNCTIONS

We consider a system of externally driven electrons and phonons described by the

Hamiltonian [16,23,26,38],

H ¼ H0 þ Hep þ Hex ð1Þ

where H0 represents the non-interacting electrons and phonons,

H0 ¼ð

drc†ðrÞh0ðrÞcðrÞ þð

drf†ðrÞV0ðrÞfðrÞ ð2Þ

h0ðrÞ ¼ ð2"2=2mÞ72 is the kinetic energy and m is the electron mass. cðc†Þ represent the

annihilation (creation) operators which satisfy the Fermi anticommutation relations,

cðrÞc†ðr0Þ þ c†ðr0ÞcðrÞ ¼ dðr 2 r0Þ

c†ðrÞc†ðr0Þ þ c†ðr0Þc†ðrÞ ¼ 0

cðrÞcðr0Þ þ cðr0ÞcðrÞ ¼ 0

ð3Þ

and fðf†Þ are boson operators with the commutation relations,

fðrÞf†ðr0Þ2 f†ðr0ÞfðrÞ ¼ dðr 2 r0Þ

f†ðrÞf†ðr0Þ2 f†ðr0Þf†ðrÞ ¼ 0

fðrÞfðr0Þ2 fðr0ÞfðrÞ ¼ 0

ð4Þ

The second term in Eq. (1) denotes the electron–phonon interaction,

Hep ¼ð

drlðrÞ½f†ðrÞ þ fðrÞ�c†ðrÞcðrÞ ð5Þ

where lðrÞ is the coupling strength. Finally, Hex represents the coupling to a time-

dependent external potential jðr; tÞ;

Hex ¼ð

drjðr; tÞc†ðrÞcðrÞ ð6Þ

We next briefly survey some properties of Liouville space superoperators that will be

useful in the following derivations [49]. The elements of the Hilbert space N £ N density

matrix, rðtÞ; are arranged as a Liouville space vector (bra or ket) of length N2: Operators

of N2 £ N2 dimension in this space are denoted as superoperators. With any Hilbert space

operator A; we associate two superoperators AL (left) and AR (right) defined through their

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action on another operator X as

ALX ; AX and ARX ; XA ð7Þ

We further define symmetric and antisymmetric combinations of these superoperators

Aþ ¼1

2ðAL þ ARÞ and A2 ¼ ðAL 2 ARÞ ð8Þ

The commutator and anticommutator operations in Hilbert space can thus be

implemented with a single multiplication by a ‘2’ and ‘þ’ superoperator, respectively.

We further introduce the Liouville space–time ordering operator T . This is a key

ingredient for extending NEGF to superoperators: when applied to a product of

superoperators it reorders them so that time increases from right to left. We define

kAðtÞl ; Tr{AðtÞreq} where req ¼ rðt ¼ 0Þ represents the equilibrium density matrix of

the interacting system. It is straightforward to see that for any two operators A and B we

have

kT AþðtÞB2ðt0Þl ¼ 0; t0 . t ð9Þ

kT AþðtÞB2ðt0Þl is thus always a retarded function. This follows from the definitions (8).

Since a ‘2’ superoperator corresponds to a commutator in Hilbert space, this implies that

for t , t0; kAþðtÞB2ðt0Þl becomes a trace of a commutator and must vanish, i.e.

kT AþðtÞB2ðt0Þl ¼ Tr{B2ðt

0ÞAþðtÞreq} t , t0

¼1

2Tr{½Bðt0Þ;AðtÞreq þ reqAðtÞ�} ¼ 0

Similarly, it follows that the trace of two ‘minus’ operators always vanishes

kT A2ðtÞB2ðt0Þl ¼ 0 for all t and t0 ð10Þ

We shall make use of Eqs. (9) and (10) for discussing the retarded and advanced Green

functions in Appendix 14D. Superoperator algebra was surveyed in Ref. [49].

In Liouville space the density matrix, r(t) is a vector whose time dependence is given by

rðtÞ ¼ Gðt; t0Þrðt0Þ ð11Þ

with the Green function,

Gðt; t0Þ ¼ T exp 2i

"

ðt

t0

H 2ðtÞdt

� �ð12Þ

and H 2 is the superoperator corresponding to the Hamiltonian (Eq. (1)). Note that unlike

Hilbert space, where time dependence of the ket and the bra is governed by forward and

backward time-evolution operators, respectively, in Liouville space one keeps track of

both bra and ket simultaneously and the density matrix needs only to be propagated

forward in time (Eq. (11)).

To introduce the interaction picture in Liouville space we partition H 2 ¼

H 02 þH 02 corresponding to the non-interacting and interaction Hamiltonians.

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With this partitioning, Eq. (12) can be written as

Gðt; t0Þ ¼ G0ðt; t0ÞGIðt; t0Þ ð13Þ

where G0 represents the time evolution with respect to H0;

G0ðt; t0Þ ¼ uðt 2 t0Þ exp 2i

"H 02ðt 2 t0Þ

� �ð14Þ

GIðt; t0Þ is the time evolution operator in the interaction picture,

GIðt; t0Þ ¼ T exp 2i

"

ðt

t0

~H 02ðtÞdt

� �ð15Þ

and ~H 02 is the interaction picture representation of H 0

2: We shall denote superoperators

in the interaction picture by a (,),

~AaðtÞ ; G†0ðt; t0ÞAaðt0ÞG0ðt; t0Þ ð16Þ

where a ¼ þ;2 or L, R. Superoperators in the Heisenberg picture will be represented

by a caret

AaðtÞ ; G†ðt; t0ÞAaðt0ÞGðt; t0Þ ð17Þ

By adiabatic switching of the interaction H 02 starting at t0 ¼ 21 we have

rðtÞ ¼ r0 2i

"

ðt

21dtG0ðt; tÞH

02ðtÞrðtÞ ð18Þ

where r0 ¼ rð21Þ is the equilibrium density matrix of the non-interacting system

r0 ¼expð2bH0Þ

Tr{expð2bH0Þ}ð19Þ

An iterative solution of Eq. (18) yields

rðtÞ ¼ GIðt;21Þr0 ð20Þ

which can also be obtained by applying the time evolution operator (15) to r(t0) and

setting t0 ¼ 21: Using Eq. (20), the equilibrium density matrix of the interacting system

can be generated from the non-interacting one by switching on the interactions

adiabatically, starting at t ¼ 21: The external potential is constant in time for t , 0 and

is assumed to be time dependent only for t . 0: We then get

req ¼ GIð0;21Þr0 ð21Þ

This adiabatic connection formula has been shown [49] to be very useful for calculating

expectation values using the interaction picture. In the corresponding Gellman-law

expression in Hilbert space [50] there is an extra denominator that takes care of the phase


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of the wavefunction. This is not necessary in Liouville space since the density matrix

does not acquire such a phase.

In the Heisenberg picture, the expectation value of an operator AaðtÞ is given by

kAaðtÞl ; Tr{AaðtÞreq} ð22Þ

where req ¼ rðt ¼ 0Þ: Using Eqs. (16), (17) and (20), this can be recast in the interaction

picture as

kAaðtÞl ¼ Tr{ ~AaðtÞGIðt;21Þr0} ; k ~AaðtÞGIðt;21Þl0 ð23Þ

Eq. (23) is a good starting point for developing a perturbation theory around the non-

interacting system. Through Eqs. (22) and (23) we also define the expectation values k· · ·land k· · ·l0. While the former represents the trace with respect to the interacting density

matrix, the latter is defined with respect to the non-interacting density matrix. This will be

used in the following.

Corresponding to the Hilbert space electron and phonon operators, c; c†; f and f† we

define ‘left’ (a ¼ L) and ‘right’ (a ¼ R) superoperators, ca; c†a; fa and f†

a: The

dynamics of a superoperator, ca; is described by the generalized Liouville equation,

2i"›caðtÞ

›t¼ ½H 2ðtÞ; caðtÞ� ¼ H 2ðtÞcaðtÞ2 caðtÞH 2ðtÞ ð24Þ

where H 2 is the superoperator corresponding to the Hamiltonian given in Eq. (1). A

similar equation can be written down for the phonon superoperators. In order to evaluate

the commutator appearing in the RHS of Eq. (1), we need the commutation relations of

superoperators [51]. The ‘left’ and the ‘right’ operators always commute. Thus, for

a – b we have

½caðrÞ;cbðr0Þ� ¼ ½c†

aðrÞ;c†bðr

0Þ� ¼ ½c†aðrÞ;cbðr

0Þ� ¼ 0

½faðrÞ;fbðr0Þ� ¼ ½f†

aðrÞ;f†bðr

0Þ� ¼ ½f†aðrÞ;fbðr

0Þ� ¼ 0

ð25Þ

For Fermi superoperators we have

caðrÞcaðr0Þ þ caðr

0ÞcaðrÞ ¼ 0

c†aðrÞc

†aðr

0Þ þ c†aðr

0Þc†aðrÞ ¼ 0

caðrÞc†aðr

0Þ þ caðr0Þc†

aðrÞ ¼ dðr 2 r0Þ

ð26Þ

Similarly for the boson operators

f†aðrÞf

†aðr

0Þ2 f†aðr

0Þf†aðrÞ ¼ 0

faðrÞfaðr0Þ2 faðr

0ÞfaðrÞ ¼ 0

faðrÞf†aðr

0Þ2 f†aðr

0ÞfaðrÞ ¼ kadðr 2 r0Þ

ð27Þ

Here ka ¼ 21 for a ¼ R and unity for a ¼ L:

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Using the commutation relations (25) and (26) and the identity

ðXY…ZÞa ¼ XaYa…Za; a ¼ L;R ð28Þ

we can recast H 2 in terms of the elementary field superoperators,

H 2 ¼ H 02 þHep2 þH

ex2 ð29Þ

with

H 02 ¼X

a¼L;R

ka

ðdrðc†

aðrÞh0ðrÞcaðrÞ þ f†aðrÞV0ðrÞfaðrÞÞ

He2p2 ¼

Xa¼L;R

ka

ðdrlðrÞFaðrÞc

†aðrÞcaðrÞ

Hex2 ¼

Xa¼L;R

ka

ðdrc†

aðrÞjðr; tÞcaðrÞ

ð30Þ

where Fa ¼ fa þ f†a:

We next define electron and phonon superoperator Green functions

Gabðrt;r0t0Þ¼2i

"kT caðr;tÞc

†bðr

0;t0Þl

Dabðrt;r0t0Þ¼2i

"kT Faðr; tÞF

†bðr

0; t0Þlð31Þ

As shown in Ref. [44] (see Appendix 14A), GLL, GRR, GLR and GRL, respectively,

coincide with the standard Hilbert space–time ordered GT ; antitime ordered G~T; lesser

G, and greater G. Green functions defined on a closed time loop.

Using the commutation relations (3), the Heisenberg equations of motion for

superoperator caðtÞ reads

i"ka›caðr; tÞ

›t¼ hðr; tÞcaðr; tÞ þ lðrÞFaðr; tÞcaðr; tÞ ð32Þ

where hðr; tÞ ¼ h0ðrÞ þ jðr; tÞ: By taking the time derivative of the electron Green

function in Eq. (31) and using Eq. (32), we obtain the equation of motion for Gab;

i"›

›t2 kahðr; tÞ

� �Gabðrt; r0; t0Þ ¼ dabdðx 2 x0Þ2

i

"kalðrÞ

� kT Faðr; tÞcaðr; tÞc†bðr

0; t0Þl ð33Þ

In order to derive the equation of motion for the phonon Green function Da; we add the

following coupling term ðdrJðr; tÞFðrÞ ð34Þ

to the Hamiltonian. Here Jðr; tÞ is some artificial field that will be set to zero at the end of

calculations. This new term does not effect the electron Green function since F and c


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commute. Thus, superoperator Hex2 in Eq. (30) takes the form

Hex2 ¼

Xa¼L;R

ka

ðdr½c†

aðrÞjðr; tÞcaðrÞ þ Jaðr; tÞFaðrÞ� ð35Þ

where fields JL and JR couple to the ‘left’ and ‘right’ boson operators, respectively. Using

the boson commutation (Eq. (26)), the Heisenberg equation for superoperator fa is

obtained as

2i"ka›faðr; tÞ

›t¼ V0ðrÞfaðr; tÞ þ lðrÞc†

aðr; tÞcaðr; tÞ þ Jaðr; tÞ ð36Þ

Using Eq. (36) we next obtain the equation of motion for operator Fa

2"2ka

V0ðrÞ

›2Faðr; tÞ

›t2¼ V0ðrÞFaðr; tÞ þ 2lðrÞc†

aðr; tÞcaðr; tÞ þ 2Jaðr; tÞ ð37Þ

Taking trace with respect to the density matrix rðt ¼ 0Þ we obtain

21

2

"2ka

V0ðrÞ

›2

›t2þV0ðrÞ

!kFaðr; tÞl ¼ lðrÞkc†

aðr; tÞcaðr; tÞlþ Jaðr; tÞ ð38Þ

Using the interaction picture representation (Eq. (23)) with Hex2 given by Eq. (35) we can

write

kFaðr; tÞl ¼ k ~Fðr; tÞGIðt;21Þl0 ð39Þ

By taking the functional derivative with respect to Jb; and setting JL ¼ JR ¼ 0; we obtain

d

dJbðr0; t0Þ

kFaðr; tÞl

JL¼JR¼0

¼ 2i

"ebkT ~Faðr; tÞ ~F†

bðr0; t0ÞGIðt;21Þl0

¼ kbDabðrt; r0t0Þ ð40Þ

Using Eqs. (38)–(40), the equation of motion for the phonon Green function is obtained as

21

2

"2kb

V0ðrÞ

›2

›t2þV0ðrÞ

!Dabðrt; r0; t0Þ ¼ dabdðx 2 x0Þ þ

i

"lðrÞka

� kT c†aðr; tÞcaðr; tÞF†

bðr0; t0Þl ð41Þ

We shall denote the space and time coordinates collectively by x ¼ r; t; thus in Eqs. (33)

and (41) dðx 2 x0Þ ; dðr 2 r0Þdðt 2 t0Þ:Following Keldysh, we shall rearrange the superoperator Green functions in a 2 £ 2

matrix �G;

�Gðx; x0Þ ¼GLLðx; x0Þ GLRðx; x0Þ

GRLðx; x0Þ GRRðx; x0Þ

!ð42Þ

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and similarly the phonon Green function matrix �D with elements Dab: The corresponding

Green functions of the non-interacting system described by the Hamiltonian (2) are

denoted by �G0 and �D0; respectively. These are given by

G0abðrt; r0t0Þ ¼ i"

›

›t2 kahðr; tÞ

� �21

dabdðx 2 x0Þ

D0abðrt; r0t0Þ ¼ 2

1

2

"2kb

V0ðrÞ

›2

›t2þV0ðrÞ

!21

dabdðx 2 x0Þ

ð43Þ

Using our matrix notation, we can recast Eqs. (33) and (41) in the form of Dyson

equations

�G ¼ �G0 þ �G0 �S �G;

�D ¼ �D0 þ �D0 �P �Dð44Þ

The effect of all interactions is now included in the electron ð �SÞ and phonon ð �PÞ self-

energies. Exact expressions for the self-energies are obtained by comparing Eqs. (33) and

(41) with Eq. (44)

Sabðrt; r0t0Þ ¼ 2i

"kalðrÞ

Xb0

ðdtð

dr1kT F†aðr; tÞcaðr; tÞc†

b0 ðr1; tÞlG21b0bðr1t; r0t0Þ

Pabðrt; r0t0Þ ¼i

"kalðrÞ

Xb0

ðdtð

dr1kT c†aðr; tÞcaðr; tÞF†

b0 ðr1; tÞlD21b0bðr1t; r0t0Þ

ð45Þ

Eqs. (43)–(45) are exact and constitute the NESGF formalism.

In order to evaluate the self-energies perturbatively, we rewrite the Green functions,

Eq. (31), in the interaction picture

Gabðrt; r0t0Þ ¼ 2i

"kT ~caðr; tÞ ~c†


Dabðrt; r0t0Þ ¼ 2i

"kT ~Fðr; tÞ ~F†


ð46Þ

where GIðt;21Þ is given by Eq. (15) with t0 ¼ 21: Using Eqs. (13), (16) and (20), the

self-energies (45) can also be expressed in the interaction picture as

Sabðrt;r0t0Þ¼2i

"kalðrÞ

Xb0

ððdtdr1kT ~F†

aðr;tÞ ~caðr;tÞ ~c†b0 ðr1;tÞGIðt;21Þl0G21

b0bðr1t;r0t0Þ

Pabðrt;r0t0Þ¼i

"kalðrÞ

Xb0

ððdtdr1kT ~c†

aðr;tÞ ~caðr;tÞ ~F†b0 ðr1;tÞGIðt;21Þl0D21

b0bðr1t;r0t0Þ

ð47Þ

Eq. (47) together with Eq. (46) constitutes closed form equations for the self-energies

where all the averages are given in the interaction picture, k· · ·l0, with respect to


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the non-interacting density matrix. By expanding GI (Eq. (15)) perturbatively in ~H 02 we

can obtain perturbative expansion for the self-energies. Each term in the expansion can

be calculated using Wick’s theorem for superoperators [49] which is given in

Appendix 14E. This results in a perturbative series in terms of the zeroth order Green

functions.

14.3 THE CALCULATION OF MOLECULAR CURRENTS

We have applied the NESGF to study the charge conductivity of a molecular wire

attached to two perfectly conducting leads. In the simplest approach the leads ‘a’ and ‘b’

are treated as two free electron reservoirs. Nuclear motions in the molecular region are

described as harmonic phonons which interact with the surrounding electronic structure

and the environment (secondary phonons) [26]. We first recast the general Hamiltonian,

Eq. (1), in a single electron local basis and partition it as

H ¼ Hf þ Hint ð48Þ

where Hf represents the free, non-interacting electrons and phonons and with no coupling

between molecule and leads

Hf ¼Xi; j

Ei; jc†i cj þ

Xk[a;b

ekc†kck þ

Xl

Vlf†l fl þ

Xm

vmf†mfm ð49Þ

The indices ði; jÞ represent the electronic basis states corresponding to the molecule, k

labels the electronic states in the leads (a and b), l denotes primary phonons which

interact with the electrons and m denotes the secondary phonons which are coupled to the

primary phonons and constitute a thermal bath. The applied external voltage V maintains

a chemical potential difference, ma 2 mb ¼ eV ; between the two leads and also modifies

the single electron energies. In addition it provides an extra termP

i Vic†i ci which is

included in the zeroth order Hamiltonian, Hf ; by modifying the single electron energies.

The interaction Hamiltonian is given by

Hint ¼X

k[a;b;i

ðVkic†kci þ h:c:Þ þ

Xl;i

lliFlc†i ci þ

Xl;m

UlmFlFm ð50Þ

The three terms represent the molecule/lead interaction, coupling of primary phonons

with the molecule and the interaction of primary and secondary phonons, respectively.

The total current passing through the junction can be expressed in terms of the electron

Green functions and the corresponding self-energies. At steady state it is given by (see

Appendix 14B, Eq. (B22))

IT ¼2e

"

Xij0

ð dv

2p

hS

ij 0

LRðvÞGj 0iRLðvÞ2 S

ij 0

RLðvÞGj 0iLRðvÞ

ið51Þ

The electron Green functions G0LR and G0

RL correspond to the free Hamiltonian, Hf ; and

the self-energies SLR and SRL represent the effects of all interactions (Eq. (50)).

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Sijab has contributions from the electron-lead (s) and electron-phonon (J ) interactions

SijabðvÞ ¼ s

ijabðvÞ þJ

ijabðvÞ ð52Þ

These are given in Eqs. (C29) and (C38). The self-energy expressions (C38) and (C40)

are calculated perturbatively to second order in the electron–phonon coupling in terms of

the zeroth order Green functions (Eq. (55)). The simplest expression for current is

obtained by substituting Eqs. (55), (C29) and (C38) in Eq. (51). This zeroth order result

can be improved by using the renormalized Green functions obtained from the self-

consistent solution of the Dyson equation (44).

In order to solve self-consistently for the electron Green functions that appear in

the current formula, Eq. (51), the self-energy is calculated under the Born

approximation by replacing the zeroth order Green functions, G0ab and D0

ab with

the corresponding renormalized Green functions, Gab and Dab; as is commonly done

in mode-coupling theories [52,53]. This approximation sums an infinite set of non-

crossing diagrams [54,55] that appear in the perturbation expansion of the many-body

Green function, Gab:Since the electron self-energy (Eq. (C38)) also depends on the phonon Green function,

the phonon self-energy, Pll0

ab; is also required for a self-consistent solution of the electron

Green functions. The phonon self-energy calculated in Appendix 14C is given by

PijabðvÞ ¼ g

ijabðvÞ þ L

ijabðvÞ ð53Þ

where gijabðvÞ (Eq. (C30)) and L

ijabðvÞ (Eq. (C40)) represent the contributions from the

phonon–phonon and the electron–phonon interactions, respectively.

Computing the renormalized electron and phonon Green functions and the

corresponding self-energies involves the self-consistent solution of the following

coupled equations for the Green functions:

GLRðvÞ ¼ G0LLðvÞSLLðvÞGLRðvÞ þ G0

LLðvÞSLRðvÞGRRðvÞ

GRLðvÞ ¼ G0RRðvÞSRLðvÞGLLðvÞ þ G0

RRðvÞSRRðvÞGRLðvÞ

GLLðvÞ ¼ G0LLðvÞ þ G0

LLðvÞSLLðvÞGLLðvÞ þ G0LLðvÞSLRðvÞGRLðvÞ

GRRðvÞ ¼ G0RRðvÞ þ G0

RRðvÞSRLðvÞGLRðvÞ þ G0RRðvÞSRRðvÞGRRðvÞ

ð54Þ

Similarly the equations for the phonon Green functions are obtained by replacing Gab

with Dab and Sab with Pab: Here Green functions corresponding to the free Hamiltonian

G0ijab and D0ll0

ab are given by

G0ijabðvÞ ¼

dab

vdij 2 kaEij þ ih; D0ll0

ab ðvÞ ¼2Vldabdll0

ka"2v2 2V2

l þ ihð55Þ


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where we set " ¼ 1 and h! 0: Eij ¼ Ei 2 Ej is the energy difference between single

electron ith and jth states. Vl denotes the molecular phonon eigenstates.

Once the Green functions, GLR; GRL and the corresponding self-energies SLR; SRL;are obtained from the self-consistent solution of Eq. (54) together with Eq. (52),

formula (51) can be used to calculate the total current through the molecular junction.

14.4 DISCUSSION

In this chapter, we have developed the NESGF formalism and applied it to the

computation of molecular current. The Liouville space–time ordering operator provides

an elegant way for performing calculations in real time, thus avoiding the artificial

backward and forward time evolution required in Hilbert space (Keldysh loop). Wick’s

theorem for superoperators is used to compute the self-energies perturbatively to the

second order in phonon–electron coupling. Eq. (54) have been derived earlier by many

authors [19,43,55]. Recently, Galperin et al. [26] have used a fully self-consistent

solution to study the influence of different interactions on molecular conductivity for a

strong electron–phonon coupling. The main aim of the present work is to demonstrate

that by doing calculations in Liouville space one can avoid the backward/forward time

evolution (Keldysh loop) required in Hilbert space. This originates from the fact that in

Liouville space both ket and bra evolve forward in time. Thus, one can couple the

system with two independent fields, ‘left’ and ‘right’. This property of Liouville space

can be used to construct real (physical) time generating functionals for the non-

perturbative calculation of the self-energies.

The present model [37–39] ignores electron–electron interactions. These may be

treated using the GW technique [56–58] formulated in terms of the superoperators and

extended to non-equilibrium situations. All non-equilibrium observables can be

obtained from a single generating functional in terms of ‘left’ and ‘right’ operators.

The retarded (advance) Green function that describes the forward (backward) motion of

the system particle can also be calculated in terms of the basic Green functions, Gab

(see Appendix 14D).

The NESGF formulation can also be recast in terms of the þ and 2 (rather than L/R)

superoperators which are more directly related to observables. This is done in

Appendix 14D. We focused on the primary quantities that are represented in terms of

the ‘left’ and ‘right’ superoperators and all other quantities are obtained as the linear

combination of these basic operators.

14.5 ACKNOWLEDGEMENTS

The support of the National Science Foundation (Grant No. CHE-0132571) and NIRT

(Grant No. EEC 0303389) is gratefully acknowledged. We wish to thank Prof. Wilson Ho

for useful discussions and Prof. Abraham Nitzan for sending us the preprint of his paper

(Ref. [23]).

Chapter 14384

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APPENDIX 14A: SUPEROPERATOR EXPRESSIONS FOR

THE KELDYSH GREEN FUNCTIONS

The standard NEGF theory formulated in terms of the four Hilbert space Green functions:

time ordered ðGT Þ; anti-time ordered ðG~TÞ; greater (G.) and lesser (G,) [23,25]. These

are defined in the Heisenberg picture as

GT ðx; x0Þ ; 2i

"kTcðxÞc†ðx0Þl

¼ 2i

"uðt 2 t0ÞkcðxÞc†ðx0Þlþ uðt0 2 tÞkc†ðx0ÞcðxÞl

G~Tðx; x0Þ ; 2

i

"k ~TcðxÞc†ðx0Þl

¼ 2i

"uðt0 2 tÞkcðxÞc†ðx0Þlþ uðt 2 t0Þkc†ðx0ÞcðxÞl

G.ðx; x0Þ ; 2i

"kcðxÞc†ðx0Þl

G,ðx; x0Þ ;i

"kc†ðx0ÞcðxÞl

ðA1Þ

These are known as T ð ~TÞ is the Hilbert space–time (anti-time) ordering operator: when

applied to a product of operators, it reorders them in ascending (descending) times from

right to left.

The four Green functions that show up naturally in Liouville space are defined as

GLLðx; x0Þ ¼ 2i

"kT cLðxÞc

†Lðx

0Þl

GRRðx; x0Þ ¼ 2i

"kT cRðxÞc

†Rðx

0Þl

GLRðx; x0Þ ¼ 2i

"kT cLðxÞc

†Rðx

0Þl

GRLðx; x0Þ ¼ 2i

"kT cRðxÞc

†Lðx

0Þl

ðA2Þ

T is the Liouville space–time ordering operator, which rearranges all superoperators in

increasing order of time from right to left.


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To establish connection between Liouville space and Hilbert space Green functions we

shall convert superoperators back to ordinary operators [44]. For GLR, and GRL, we obtain

GLRðx; x0Þ ; 2i

"Tr{T cLðxÞc

†Rðx

0Þreq}

¼ 2i

"Tr{cðxÞreqc

†ðx0Þ}

¼ 2i

"kc†ðx0ÞcðxÞl ¼ G,ðx; x0Þ

GRLðx; x0Þ ; 2i

"Tr{T cRðxÞc

†Lðx

0Þreq}

¼ 2i

"Tr{c†ðx0ÞreqcðxÞ}

¼ 2i

"kcðxÞc†ðx0Þl ¼ G.ðx; x0Þ

ðA3Þ

where req is the fully interacting many body equilibrium density matrix.

For GLL and GRR we have two cases

(i) For t . t0; we get

GLLðx; x0Þ ; 2i

"Tr{T cLðxÞc

†Lðx

0Þreq}

¼ 2i

"Tr{cðxÞc†ðx0Þreq} ¼ 2

i

"kcðxÞc†ðx0Þl

GRRðx; x0Þ ; 2i

"Tr{T cRðxÞc

†Rðx

0Þreq}

¼ 2i

"Tr{reqc

†ðx0ÞcðxÞ} ¼ 2i

"kc†ðx0ÞcðxÞl

ðA4Þ

(ii) For the reverse case, t , t0; we get

GLLðx; x0Þ ; 2i

"Tr{T cLðxÞc

†Lðx

0Þreq}

¼ 2i

"Tr{c†ðx0ÞcðxÞreq} ¼ 2

i

"kc†ðx0ÞcðxÞl

GRRðx; x0Þ ; 2i

"Tr{T cRðxÞc

†Rðx

0Þreq}

¼ 2i

"Tr{reqcðxÞc

†ðx0Þ} ¼ 2i

"kcðxÞc†ðx0Þl

ðA5Þ

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Combining Eqs. (A4) and (A5) we can write

GLLðx; x0Þ ; 2i

"Tr{T cLðxÞc

†Lðx

0Þreq}

¼ 2i

"½uðt 2 t0ÞkcðxÞc†ðx0Þl2 uðt0 2 tÞkc†ðx0ÞcðxÞl�

¼ GT ðx; x0Þ

GRRðx; x0Þ ; 2i

"Tr{T cRðxÞc

†Rðx

0Þreq}

¼ 2i

"½uðt 2 t0Þkc†ðx0ÞcðxÞl2 uðt0 2 tÞkcðxÞc†ðx0Þl�

¼ G~Tðx; x0Þ

ðA6Þ

Eqs. (A3) and (A6) establish the equivalence of Hilbert and Liouville space Green

functions and they can be summarized as

GLLðx; x0Þ ¼ GT ðx; x0Þ; GRRðx; x0Þ ¼ G�Tðx; x0Þ

GLRðx; x0Þ ¼ G,ðx; x0Þ; GRLðx; x0Þ ¼ G.ðx; x0ÞðA7Þ

APPENDIX 14 B: SUPEROPERATOR GREEN FUNCTION

EXPRESSION FOR THE CURRENT

In this appendix, we present a formal microscopic derivation for the current flowing

through a conductor. The conductor could be a molecule or a metal or any conducting

material attached to two electrodes held at two different potentials.

In Hilbert space the charge current–density is given by

jðr; tÞ ¼ 2ie"

2mk½c†ðr; tÞ7cðr; tÞ2 ð7c†ðr; tÞÞcðr; tÞ�l ðB8Þ

where e and m are the electron charge and mass, respectively. Eq. (B8) can also be

expressed in a slightly modified form as

jðr; tÞ ¼ie"

2m½kð72 70Þc†ðr; tÞcðr0; t0Þl�x0¼x ðB9Þ

where 70 represents the derivative with respect to r0.

Using relation (A7) the current density can be expressed in terms of the superoperator

Green function as

jðr; tÞ ¼ 2e"2

2m½ð72 70ÞGLRðrt; r0t0Þ�x0¼x ðB10Þ


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At steady state, the Green functions only depend on the time difference ðt 2 t0Þ and the

total current density (JT) becomes time independent. Transforming to the frequency

(energy) domain, the current density per unit energy is

jðr;EÞ ¼ 2e"

2m½ð72 70ÞGLRðrr0;EÞ�r0¼r ðB11Þ

and the total current density

JTðrÞ ¼ð dE

2pjðr;EÞ ðB12Þ

Eq. (B12) provides a recipe for calculating the current profile across the conductor once

the Green function GLR is known from the self-consistent solution of the Dyson equation.

For computing the total current passing through the conductor, Eq. (B11) can be

expressed in the form of Eq. (51). In order to get the total current per unit energy ðITðEÞÞ

passing between electrode and conductor we need to integrate the current density over the

surface area of the conductor–electrode contact

ITðEÞ ¼ð

sjðr;EÞ · ndS ¼

ð7 · jðr;EÞdr ðB13Þ

where n is the unit vector normal to surface S: Substituting into Eq. (B13) from Eq. (B11),

we get

ITðEÞ ¼ 2e"

2mTr½ð72 2 702ÞGLRðrr0;EÞ� ðB14Þ

In general, a conductor–electrode system can be described by the Hamiltonian

H ¼ H0 þ Hint ðB15Þ

where H0 represents the non-interacting part

H0 ¼ð

drc†ðrÞh0ðrÞcðrÞ ðB16Þ

where h0ðrÞ ¼ 2 "2

2m72 and all the interaction terms (conductor–electrode, electron–

phonon) are included in Hint: The total current per unit energy, Eq. (B14), is

ITðEÞ ¼ 2e

"Tr½ðh0ðrÞ2 hp

0ðr0ÞÞGLRðrr0;EÞ� ðB17Þ

The Dyson equations for the retarded Green function (see Appendix 14D, Eq. (D48)) in

frequency (energy) can be expressed in the matrix form as

h0Gr ¼ EGr 2 I 2 srGr ðB18Þ

where I is the identity matrix and Sr is the retarded self-energy, Eq. (D50). E ¼ "v is a

number. Henceforth, we write all the expressions in the matrix notation. Taking

Chapter 14388

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the complex conjugate of Eq. (B18), we obtain the Dyson equation for the advanced

Green function,

Gah0 ¼ EGa 2 I 2 Gasa ðB19Þ

with the corresponding advanced self-energy, Sa: From the matrix Dyson equation

(D48), we also have the relation

GLR ¼ GrSLRGa ðB20Þ

Using the relations (B18)–(B20), it is easy to see that

h0GLR 2 GLRh0 ¼ GLRSa þ GrSLR 2 SrGLR 2 SLRGa ðB21Þ

Substituting this in Eq. (B17), the total current per unit energy becomes

ITðEÞ ¼2e

"Tr½SLRðEÞGRLðEÞ2 SRLðEÞGLRðEÞ� ðB22Þ

where a factor of 2 is introduced to account for the spin degeneracy.

We have calculated the total current in real space. In practice, the Green functions and

the self-energy matrices are calculated in an electronic basis ði; jÞ: The total current

through the conductor is obtained by integrating Eq. (B22) over energy resulting in

Eq. (51).

APPENDIX 14C: SELF-ENERGIES FOR SUPEROPERATOR GREEN

FUNCTIONS

The basic quantities required for describing the coupled molecule–lead system are the

one particle electron and the phonon Green functions. Following the steps outlined in

Section 14.2, the time development for various superoperators (Heisenberg equations) is

(all primed indices should be summed over)

i"ka›

›tciaðtÞ ¼ Eij0 cj0aðtÞ þ Vk0ick0aðtÞ þ ll0ifl0aðtÞciaðtÞ

2i"ka›

›tflaðtÞ ¼ li0lc

†i0aðtÞci0aðtÞ þVlflaðtÞ þ Ulm0fm0aðtÞ

i"ka›

›tckaðtÞ ¼ ekckaðtÞ þ Vki0 ci0 ðtÞ

2 i"ka›

›tfmaðtÞ ¼ vmfmaðtÞ þ Ul0mfl0 ðtÞ

ðC23Þ

Using Eq. (C23) and following the procedure described in Section 14.2, it is

straightforward to write the matrix Dyson equation (44) for the electron and phonon


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Green functions defined as

Gijabðx;x

0Þ¼2i

"kT ciaðr;tÞc

†jbðr

0; t0Þl

Dll0

abðx;x0Þ¼2

i

"kT Flaðr; tÞF

†l0bðr0;t0Þl

ðC24Þ

with the corresponding self-energy matrix elements

Sijabðt; t

0Þ ¼2i

"kaXb0; j0

ðdt

Xl0

ll0ikT Fl0aðtÞciaðtÞc†j0b0 ðtÞlþ

Xk0

Vk0ikT ck0aðtÞc†j0b0 ðtÞl

" #

�Gj0j21

b0bðt; t0Þ;J

ijabðt; t

0Þþsijabðt; t

0Þ

Pll0

abðt; t0Þ ¼

i

"

ðdtXb0;l00

Xm0

Ulm0kT Fm0aðtÞF†l00b0 ðtÞlþ

Xi0

lli0kT c†i0aðtÞci0aðtÞF

†l00b0 ðtÞl

" #

£Dl00l021b0b ðt; t0Þ; gll0

abðt; t0ÞþLll0

abðt; t0Þ ðC25Þ

The two terms in the electron self-energy represent the contributions from the phonon–

electron (J) and molecule–lead (s) interactions. Similarly, the phonon self-energy has

contributions from the electron–phonon (L) and the primary–secondary phonon (g)

couplings. The self-energy due to the molecule–lead coupling can be calculated exactly.

To this end we need to obtain the quantity kT ~ck0aðtÞc†j0b0 ðtÞl: By multiplying the third

equation in Eq. (C23) by c†j0b0 ðtÞ from the left and from the right, taking trace and

subtracting, we get (here primed indices are not summed over)

i"ka›

›t2 ek0

� �kT ck0aðtÞc

†j0b0 ðtÞl¼

Xi0

Vk0i0 kT ci0aðtÞc†j0b0 ðtÞl

) kT ck0aðtÞc†j0b0 ðtÞl¼ i"

Xi0

Vk0i0gk0 ðtÞGi0j0

ab0 ðt;tÞðC26Þ

where gkðtÞ ¼ ði"ka››t2 ek0 Þ

21: Substituting expression (C26) in Eq. (C25) gives for the

molecule–lead self-energy

sijabðt; t

0Þ ¼kadabX

k0[a;b

Vk0iVk0jgk0 ðtÞdðt2 t0Þ ðC27Þ

Similarly, the contribution to the phonon self-energy from the interaction with secondary

phonons can be calculated exactly

gll 0

ab ¼2kadabXm0

Ulm0Ul0m0g0m0 ðtÞdðt2 t0Þ ðC28Þ

where g0m0 ðtÞ ¼ ði"ka

››tþvm0 Þ21: At steady state all Green functions and self-energies

depend only on the time difference ðt1 2 t2Þ and it is very convenient to express them in

the frequency space. The self-energy contributions due to molecule–lead ðsijabÞ and

phonon–phonon ðg l0

abÞ interactions, Eqs. (C27) and (C28), can be represented in

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frequency space as

sijabðvÞ ¼ kadab

Xk0

Vk0iVk0j

kav2 ek0 þ ihðC29Þ

g ll0

abðvÞ ¼2kadabXm0

Ulm0Ul0m0

kavþvm0 þ ihðC30Þ

where h!0: However, in real calculations it is a common practice to calculate self-

energies sab and gab in the wide band approximation, implying that the real parts of the

self-energies can be ignored and the imaginary parts are considered as frequency

independent. Eqs. (C29) and (C30) then reduce to simpler forms

sijab ¼ kadab

i

2G ij; g ll0

ab ¼2kadabi

2~G ll0 ðC31Þ

where G ijð2pP

k0 Vk0iVk0j and ~G ll0 ¼ 2pP

m0 Um0lVm0l0 :The phonon contribution to the electronic self-energy is obtained perturbatively in the

phonon–electron coupling. We recast the phonon contribution (first term on the RHS of

Eq. (C25) for Sijab) in the interaction picture by writing

kT FlaðtÞciaðtÞc†jbðt

0Þl ¼ kT ~FlaðtÞ ~ciaðtÞ ~c†jbðt

0ÞGIðt;21Þl0 ðC32Þ

where

GIðt;21Þ ¼ exp 2i

"

ðdtXi0a0

ka0

Xl0

ll0i0~Fl0a0 ðtÞ ~c†

i0a0 ðtÞ ~ci0a0 ðtÞ

"(

þXk0

Vk0i0 ð ~c†k0a0 ðtÞ ~ci0a0 ðtÞþ ~c†

i0a0 ðtÞ ~ck0a0 ðtÞÞ

# )ðC33Þ

Substituting Eq. (C33) in Eq. (C32), expanding the exponential to first order in lli and

using Wick’s theorem for superoperators [49] we obtain

kT FlaðtÞciaðtÞc†jbðt

0Þl¼2"2Xl0i0a0

ka0ll0i0

ðdtD0ll0

aa0 ðt;tÞhG

0ijabðt; t

0ÞG0i0i0

a0a0 ðt;tþÞ

þG0ii0

aa0 ðt;tÞG0i0j

a0bðt; t0Þ

iðC34Þ

Here the superscript ‘0’ represents the trace with respect to the non-interacting density

matrix. The zeroth order Green functions are given in Eq. (55). The terms coming from

the lead-molecule coupling (Vki) vanish because they are odd in creation and annihilation

operators. Substituting Eq. (C34) in Eq. (C25) gives for the phonon contribution


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to the self-energy

Jijabðt; t

0Þ ¼ i"Xl1l2

kall1i kbll2jD0l1l2ab ðt; t0ÞG0ij

abðt; t0Þ

h

þ dijdabdðt2 t0ÞXi1a

0

ll2i1ka0

ðdtD

0l1l2aa0 ðt;tÞG

0i1i1a0a0 ðt;t

þÞi

ðC35Þ

In the derivation of Eq. (C35), we have used the identity

ðdtXa0j0

G0ij0

ab0 ðt;tÞG21 0j0j

b0bðt; t0Þ ¼ dabdijdðt2 t0Þ ðC36Þ

Similarly the contribution of the electron–phonon interaction to the phonon self-energy

(second term in Eq. (C25) for Pijab) can be obtained perturbatively. To the second order in

phonon–electron coupling, we obtain

Lll0

abðt; t0Þ ¼2i"

Xij

kakbllill0j½G0jibaðt

0; tÞG0ijabðt; t

0Þ þG0iiaaðt; t

þÞG0jjbbðt

0; t0þÞ� ðC37Þ

To second order in electron–phonon coupling, the electronic self-energy depends on both

the electron and phonon green functions while the phonon self-energy contains only the

electron Green functions.

At steady state we shift to the frequency domain and obtain

JijabðvÞ ¼ i"

Xl1l2

kakbll1ill2j

ð dv0

2pD

0l1l2ab ðv0ÞG

0ijabðv2 v0Þ

þ dijdabX

l1;l2;i1;a0

ka0ll1ill2i1r0

i1i1D

0l1l2aa0 ðv ¼ 0Þ ðC38Þ

where

r0ii ; i"Gii

aaðt ¼ 0Þ ¼ ið dE

2pGii

aaðEÞ ðC39Þ

The phonon self-energy becomes

Lll0

abðvÞ ¼ 2i"X

ij

kakbllill0j

ð dv0

2pG

0ijabðv

0ÞG0jibaðv

0 2 vÞ

þi

"

Xij

kaebllill0jr0iir

0jjdðv ¼ 0Þ ðC40Þ

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APPENDIX 14D: DYSON EQUATIONS IN THE 1 /2 REPRESENTATION

In this appendix, we define the retarded and advance Green’s functions and the

corresponding self-energies and relate them to the basic Green functions and self-

energies obtained in Appendix 14C. From definition (9), the Liouville space retarded (Gr)

and advance (Ga) Green functions are defined as

Gijr ðt; t0Þ ; 2

i

"kT ciþðtÞc

†j2ðt

0Þl ðD41Þ

Gija ðt; t0Þ ; 2

i

"kT ci2ðt

0Þc†jþðtÞl ðD42Þ

We further introduce the correlation function

Gijc ðt; t0Þ ; 2

2i

"kT ciþðt

0Þc†jþðtÞl ðD43Þ

It follows from Eq. (10) that there are only three Green functions in the þ /2

representation. These are given by Eqs. (D41)–(D43). Using Eq. (8) these can be

represented in terms of the basic Green functions (31) as

Gijr ðt; t0Þ ¼

1

2½G

ijLLðt; t0Þ2 G

ijLRðt; t0Þ þ G

ijRLðt; t0Þ2 G

ijRRðt; t0Þ�

¼ GijLLðt; t0Þ2 G

ijLRðt; t0Þ

Gija ðt; t0Þ ¼

1

2½G

ijLLðt; t0Þ2 G

ijRRðt; t0Þ2 G

ijRLðt; t0Þ þ G

ijLRðt; t0Þ�

¼ 2GijRRðt; t0Þ þ G

ijLRðt; t0Þ ¼ G

ijLLðt; t0Þ2 G

ijRLðt; t0Þ

Gijc ðt; t0Þ ¼

1

2½G

ijLLðt; t0Þ þ G

ijRRðt; t0Þ þ G

ijLRðt; t0Þ þ G

ijRLðt; t0Þ�

¼ GijLLðt; t0Þ þ G

ijRRðt; t0Þ

ðD44Þ

where we have used the identity GLL þ GRR ¼ GLR þ GRL which can be verified using

Eq. (10). A Dyson equation corresponding to Gr; Ga and Gc can be obtained from Eq. (44)

using unitary transformation

G ¼ S �GS21 ðD45Þ

where G represents the matrix

G ¼0 Ga

Gr Gc

!ðD46Þ

and

S ¼1ffiffi2

p1 21

1 1

!ðD47Þ


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The transformed Dyson equation (44) reads

G ¼ G0 þ G0 ~SG ðD48Þ

and the corresponding self-energy matrix reduces to

~S ¼Sc Sr

Sa 0

!ðD49Þ

with the matrix elements given by

Sijr ðt; t

0Þ ¼ SijLLðt; t0Þ þ S

ijLRðt; t0Þ

Sija ðt; t

0Þ ¼ SijRRðt; t0Þ þ S

ijLRðt; t

0Þ

Sijc ðt; t

0Þ ¼ SijRRðt; t0Þ þ S

ijLLðt; t0Þ

ðD50Þ

Similar relations also hold for the phonon Green functions and self-energies.

Using Eqs. (D44) and (D50), the retarded self-energies for electron and phonon Green

functions (retarded) coming from the electron–phonon coupling are obtained as

Jijr ðvÞ ¼ i"

Xll0

ll0illj

ð dv0

2p½D0ll0

r ðv0ÞG0ijr ðv2 v0Þ þ D0ll0

r ðv0ÞG0ijLRðv2 v0Þ

þ D0ll0

LRðv0ÞG0ij

r ðv2 v0Þ�

Lll0

r ðvÞ ¼ 2i"X

ijllill0j

ð dv0

2p

hG

0ijLRðv

0ÞG0ija ðv2 v0Þ

þ G0ijr ðv0ÞðG

0ijRLðv2 v0Þ þ Gaðv2 v0ÞÞ

iðD51Þ

Similarly the retarded self-energies due to the lead and secondary phonons can be written

in the wide band limit as

s ijr ¼

i

2G ij and g ll0

r ¼ 2i

2~G ll0 ðD52Þ

where Gij includes contributions from both the leads a and b, i.e. G ij ¼ Gija þ G

ijb :

APPENDIX 14E: WICK’S THEOREM FOR SUPEROPERATORS

Wick’s theorem for superoperators was formulated in Ref. [49]. Using Eqs. (8) and (27),

it can be shown that similar to the L and R superoperators, the commutator of ‘þ ’ and

‘2 ’ boson superoperators are also numbers. Thus, boson superoperators follow Gaussian

statistics and Wick’s theorem holds for both the L, R and ‘þ ’, ‘2 ’ representations.

However, for Fermi superoperators life is more complicated. The anticommutator

corresponding to only the ‘left’ or the ‘right’ Fermi superoperators are numbers but that

for the ‘left’ and ‘right’ superoperators, in general, is not a number. Thus, the Fermi

superoperators are not Gaussian. However, since the left and right superoperators always

commute, the following Wick’s theorem [49] can be applied to the time-ordered product

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of any number of ‘left’ and ‘right’ superoperators, e.g.

kT Ai1n1ðt1ÞAi2n2

ðt2Þ· · ·AinnnðtnÞl0 ¼

Xp

kT AianaðtaÞAibnb

ðtbÞl0 · · · kT AipnpðtpÞAiqnq

ðtqÞl0ðE53Þ

Here Ainnn; nn ¼ L;R; represents either a boson or a fermion superoperator. iana…iqvq is a

permutation of i1n1…innn and sum on p runs over all possible permutations, keeping the

time ordering. In case of fermions, each term should be multiplied by ð21ÞP; where P is

the number of permutations of superoperators required to put them into a particular order.

Only permutations among either ‘left’ or among ‘right’ superoperators count in P:The permutations among ‘L’ and ‘R’ operators leave the product unchanged.

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Author QueriesJOB NUMBER: 9585

JOURNAL: Superoperator many-body theory of molecular currents: non-equilibrium

Green functions in real time

Q1 Please check sense of the sentence ‘At equilibrium suffice it to know only the….

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Documents

Superoperator many-body theory of molecular currents: non ...mukamel.ps.uci.edu/publications/pdfs/509.pdf · ex ¼ ð drjðr;tÞc†ðrÞcðrÞð6Þ We next brieﬂy survey some properties