tomomi shimazaki et al- a theoretical study of molecular conduction. iii. a...

8/3/2019 Tomomi Shimazaki et al- A Theoretical Study of Molecular Conduction. III. a Nonequilibrium-Greensfunction

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A theoretical study of molecular conduction. III. A nonequilibrium-Greens-function-based Hartree-Fock approach

Tomomi ShimazakiDepartment of Chemical System Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo,Tokyo 113-8656, Japan

Yongqiang XueDepartment of Chemistry, Northwestern University, Evanston, Ilinois 60208 and College of Nanoscale

Science and Engineering, University at Albany, 255 Fuller Road, Albany, New York 12203

Mark A. RatnerDepartment of Chemistry, Northwestern University, Evanston, Ilinois 60208

Koichi Yamashitaa

Department of Chemical System Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo,Tokyo 113-8656, Japan

Received 6 September 2005; accepted 24 January 2006; published online 20 March 2006

Many recent experimental and theoretical studies have paid attention to the conductivity of singlemolecule transport junctions, both because it is fundamentally important and because of itssignificance in the development of molecular-based electronics. In this paper, we discuss anonequilibrium Greens function NEGF-based Hartree-Fock HF approach; the NEGF method

can appropriately accommodate charge distributions in molecules connected to electrodes. Inaddition, we show that a NEGF-based density matrix can reduce to an ordinary HF density matrixfor an isolated molecule if the molecule does not interact with electrodes. This feature of theNEGF-based density matrix also means that NEGF-based Mulliken charges can be reduced toordinary Mulliken charges in those cases. Therefore, the NEGF-based HF approach can directlycompare molecules that are connected to electrodes with isolated ones, and is useful in investigatingcomplicated features of molecular conduction. We also calculated the transmission probability andconduction for benzenedithiol under finite electrode biases. The coupling between the electrodes andmolecule causes electron transfer from the molecule to the electrodes, and the applied bias modifiesthis electron transfer. In addition, we found that the molecule responds capacitively to the appliedbias, by shifting the molecular orbital energies. 2006 American Institute of Physics.DOI: 10.1063/1.2177652

I. INTRODUCTION

There have recently been many theoretical and experi-mental studies on the electronic properties of a moleculeconnected between electrodes in order to develop molecular-based electronic circuits and devices. Although the directmeasurement of molecular conduction remains challenging,novel methods such as mechanical controlled break junction,the cross-wire tunnel junction technique, and measurementson single atomic monolayers SAMs by scanning tunnelingelectron microscopy STM can observe some features ofelectron transport through a single molecule.18 On the otherhand, some studies have reported the rectification andswitching properties in molecular conduction, and tried todevelop molecular devices such as diodes, transistors, andmemory cells.915

In theoretical studies, the transmission probability forelectron tunneling between electrodes through a single mol-ecule is usually investigated based on Greens function meth-ods, and the effects of semi-infinite electrodes are accommo-

dated as the self-energy.1618 The semi-infinite nature ofelectrodes cannot be neglected in investigating molecularconduction because of the strong modulation of the molecu-lar orbitals. There are two main modulations: the molecularorbital energies are shifted and their levels arebroadened.1921 The electron can transfer from one electrodeto the other through such broadened molecular orbitals, sothe self-energy of semi-infinite electrodes should be appro-priately accommodated. We have to obtain the surfaceGreens function of semi-infinite electrodes in order to deter-mine the self-energy. Although recursion methods have so

far been used in order to obtain the surface Greensfunction,2224 Sanvito et al. and Krstic et al. recently reporteda novel technique that can construct the surface Greensfunction from Greens functions with periodic boundaryconditions.25,26 Unlike the recursion methods, their tech-niques do not require iterative calculations for numericalconvergence and can properly accommodate the evanescentstate in molecular conduction.

The nonequilibrium Greens function NEGF methodhas recently attracted much attention because it can studymolecular conduction under finite biases.2737 The NEGFaElectronic mail: [email protected]

THE JOURNAL OF CHEMICAL PHYSICS 124, 114708 2006

0021-9606/2006/12411 /114708/12/$23.00 2006 American Institute of Physics124, 114708-1

Downloaded 28 May 2009 to 129.105.215.213. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
http://dx.doi.org/10.1063/1.2177652http://dx.doi.org/10.1063/1.2177652http://dx.doi.org/10.1063/1.2177652http://dx.doi.org/10.1063/1.2177652http://dx.doi.org/10.1063/1.2177652http://dx.doi.org/10.1063/1.2177652


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method can give the density matrix and charge distributionon the molecule even in those situations, and the electrostaticpotential can be determined from the NEGF-based chargedistribution. In many studies, the potential is used in theframework of density function theory DFT in order to ob-tain electronic structures for a molecule connected to elec-trodes with finite biases. Although this NEGF-based DFTapproach successfully brings deep insights into molecular

conduction, we discuss in this paper a NEGF-based Hartree-Fock HF approach both because the theoretical clarity inthe HF method enables us to investigate the features of mo-lecular conduction and because correlation effects might sub-sequently be added. Galperin and Nitzan recently reported aNEGF-based HF approach.38 Although they used a constantand energy-independent self-energy term, we adopt moregeneral formalisms for the surface Greens function of semi-infinite electrodes. Other HF-based approaches to transportwere reported by Mujica et al.39 and by Delaney and Greer.40

We also demonstrate a technique to obtain the NEGF-baseddensity matrix. Our approach to the NEGF-based densitymatrix is based on a spectrum representation, which is dif-

ferent from previous other studies with an inverse matrix ofeffective Hamiltonian. The spectrum representation ofGreens function can easily show a relationship between mo-lecular conduction and ordinary quantum chemical proper-ties.

II. THEORY

A. Transmission probability

The retarded self-energy R can be determined from thesurface Greens function gsurface

R of electrodes and the elec-tronic coupling matrix between the electrodes and the mol-ecule as follows:16

R = TgsurfaceR . 1

Moreover, we can use the following relationship:

A = R, 2

and R = RT if gsurfaceR = gsurface

R T. We adopt the followingequations as the surface Greens function gsurface

R to determinethe self-energy of the electrodes:16,18

gsurfaceR =

expikc

, 3a

E= + 2coskc . 3b

Here, E is the energy of the electron wave in the electrode, is the diagonal element of the unit cell, is the nearestneighbor electronic coupling element, c is the lattice con-stant, and k is the wave number. Although the above equa-tions have very simple forms, the surface Greens functionderived by Sanvito et al. and Krstic et al., which can accom-modate more complex systems, reduces to Eq. 3a in one-dimensional problems.21 Therefore, we can investigate thefundamental features even if Eq. 3a is adopted as a primi-tive model for the surface Greens function. The effectiveHamiltonian, which represents the system including a mol-ecule and semi-infinite electrodes, is determined by adding

the self-energy to the isolated molecular Hamiltonian Hmol inone-electron orbital approximation, i.e., the Fock matrix, andthe retarded and advanced Greens functions can be obtainedas follows:

H= Hmol + pR + q

R, 4a

ES HGR = ES Hmol pR q

RGR = I, 4b

GR = ES Hmol pR qR1 , 4c

GA = GR. 4d

Here, the subscripts p and q represent leads p and q,respectively. The transmission probability Tpq can be calcu-lated from the following equation see Appendix A:

Tpq = TrpGRqG

A. 5

Here, p = ipR p

A and q = iqR q

A. The current Ipthrough a molecule can be determined from the transmissionprobability and Landauer formula by using the followingequations when the voltage V is applied to the molecule, as

shown in Fig. 1:41,42

Ip =2e

h

+

TEfqE p fpE qdE, 6a

p = EF eV, 6b

q = EF+ 1 eV. 6c

Here, EF is the Fermi energy of the electrode at zerobias, is the chemical potential, and is a parameter thatvaries from 0 to 1; we set =0.5.

B. The nonequilibrium Greens function

The molecular orbitals that interact with the semi-infiniteelectrodes are broadened because of interaction with theelectronic states of the electrodes, and the electron transfersfrom one electrode to the other along such broadened mo-lecular orbitals. On the other hand, we can control the elec-tron occupation of the broadened molecular orbitals by ap-plying a bias to the electrodes because the bias can changethe chemical potential of the electrodes; the electron occupa-tion number of the broadened molecular orbitals is not al-ways an integer, unlike the situation in the isolated molecule,as seen in Fig. 2. Moreover, the occupation becomes very

FIG. 1. Molecular conduction between leads p and q.

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complicated when different biases are applied to each elec-trode. The NEGF methods have been widely studied to ac-commodate these problems, and the density matrix for mol-ecules connected to electrodes can be calculated. In thefollowing, the density matrix of a molecule under a finitebias is derived based on the NEGF method.

In the NEGF formalism, not only the retarded and ad-vanced Greens functions but also the following lesser andgreater Greens functions must be accommodatedtogether:31,34,4345

GRx1,x2 = it1 t2x1,x2 , 7a

GA

x

1,x

2 = +i

t2

t1

x

1,

x

2 , 7b

Gx1,x2 = + ix2x1 , 7c

Gx1,x2 = ix1x2 . 7d

Here, x1 = r1 , t1 and x2 =r2 , t2. These four Greensfunctions are not independent and have the following rela-tionships:

GRx1,x2 =t1 t2Gx1,x2 G

x1,x2 , 8a

GAx1,x2 =t2 t1Gx1,x2 G

x1,x2 , 8b

and therefore

GRx1,x2 GAx1,x2 = G

x1,x2 Gx1,x2. 9

The charge density of the electron and the hole can bedetermined by the lesser and greater Greens functions, re-spectively,

nex1 = x1x1 = iG

x1,x1 , 10a

nhx1 = x1x1 = iG

x1,x1. 10b

The density matrix can be calculated from the lesserGreens function as follows:

Dr1,r2 =1

2i

+

Gr1,r2;EdE, 11a

Gr1,r2;E =

+

expiEtGx1,x2dt, t= t2 t1.

11b

In order to numerically calculate the density matrix, weneed the following steps. First, define the spectrum functionAr1 , r2 ;E as follows:

Ar1,r2;E =

+

expiEtAx1,x2dt, t= t2 t1 ,

12a

Ax1,x2 = iGRx1,x2 G

Ax1,x2

= iGx1,x2 Gx1,x2 . 12b

We assume that the following relation is applicable fornonequilibrium as well as equilibrium problems:

Gr1,r2;E = i fEAr1,r2;E . 13

Here, fE is the Fermi-Dirac distribution function. Wethen obtain

G = i fEA = i fEiGR GA

= i fEiGAGA1 GR1GR

= i fEiGAEs Hmol A Es Hmol

RGR

= i fEGAGR, 14

and similarly,

G

= i fEGR

GA

. 15Finally, the density matrix is given as

D =1

2

+

fEGAGRdE

=1

2

+

fpEGApG

R + fqEGAqG

RdE. 16

Now we can clearly distinguish the contributions fromleads p and q in the density matrix under the condition of afinite bias on a molecule.

C. Analysis of the NEGF-based density matrix by anapproximate Greens function

Equations 14 and 15 can also be expressed as fol-lows:

G = i fEA = i fEiGR GA = 2i fEIm GR. 17

Therefore, another expression for the density matrix canbe obtained as follows:

D = 1

+

fEIm GRdE. 18

FIG. 2. Electron occupation of orbitals in a molecule connected to semi-infinite electrodes.

114708-3 Hartree-Fock study of conduction J. Chem. Phys. 124, 114708 2006



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In order to analyze Eq. 18 we derive an approximateGreens function that includes the first-order energy correc-tion based on perturbation theory,20,21

Hmol + Ra

0 + a1 +

= a0 + a

1 + a0 + a

1 + , 19a

Hmola0 = a

0a

0, 19b

a1 = a

0Ra0. 19c

We remark that is the one-electron orbital. Then the fol-lowing approximate Greens function from the above equa-tions is obtained:

GRr1,r2 = a

a0r1a

0*r2

E a0 Re a

1 i Im a1 . 20

The matrix representation of the approximate Greensfunction and approximate density matrix can be expressed bythe linear combination of atomic orbital LCAO approxima-

tion of molecular orbitals a0

=C,a0

, where are theatomic orbitals,

GR =

a

C,a0

C,a0

E a0 Re a

1 i Im a1 , 21

D=

+

fEa

C,a0 C

,a0

1

2

a

E a0 Re a

12 + a/22 dE. 22

In order to obtain Eq. 22, we used

Im a1 = a

0Im Ra0 = 1

2a

0a0 = 1

2a.

23

If we assume that there is no interaction between themolecule and the electrode, the first-order energy correctionin Eq. 22 becomes zero, that is, Re

a

10 and Im

a

1

0 when R0,

limr0

1

2

E 2 + /22 E . 24

Then, the NEGF-based density matrix reduces to the or-

dinary density matrix that is used in the usual Hartree-Fockmethod,

D= limR0

1

fEIm GR dE

=

+

a

occ

C,a0

C,a0E a

0dE. 25

In order to investigate the properties of the NEGF-baseddensity matrix, we substitute the approximate Greens func-tions in Eq. 16, and we use the following equations:

p,a = a0pa

0 , 26a

q,a = a0qa

0 . 26b

Here, a =p,a +q,a. Then the approximate density matrixbecomes see also Appendix B

D =1

2a

fpEp,ap,a + q,a

+fqEq,ap,q + q,a

Im gaRdE, 27where

GR = a

gaR, 28a

gaR=

C,a0 C

,a0

E a0 Re a

1 i Im a1 , 28b

and note

Im a1 =

1

2a =

1

2p,a

1

2q,a. 29

An electron from lead p or from lead q occupies molecular

orbital a according to the ratios p,a/p,a +q,a andq,a/p,a +q,a, respectively.7 These relations show that the

electron in molecular orbital a has no scattering interactionswith phonons or other electrons; in other words, Eq. 27rep-resents a charge distribution for ballistic electron transport.Therefore, the spectrum representation of effective Greensfunction can easily show a relationship between pq,a andthe NEGF-based density matrix unlike the inverse matrix ofeffective Hamiltonian. Moreover, we remark that the ap-proximated NEGF-based density matrix Eqs. 22 and 27can well reproduce the exact one.

D. The NEGF-based density matrix

Next, we accommodate the exact Greens function byspectrum representation in order to analyze the density ma-trix given by Eq. 16. The effective Hamiltonian, in whichthe self-energy is added to the analysis of the isolated mo-lecular Hamiltonian Fock matrix, is not Hermitian, so wecannot use ordinary orthogonal relations. However, we canuse the following orthogonal relations by considering a dualspace:16,46,47

Hmol + Ra = aa, 30a

Hmol + Aa = a*a, 30b

ara*rdr= ab, 30cand therefore the exact spectrum representation of theGreens function,

GRr1,r2 = a

ar1a*r2

E a, 31

where a and a are expanded by the LCAO approximation,a = C,a

R, and a = C,a

A.




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Now Eqs. 30a and 30b can be written as follows:

Hmol + RCa

R = aSCaR, 32a

CaR = C1,a

RC2,a

R Cn,a

R T, 32b

Hmol + ACa

A = a*SCa

A, 32c

CaA = C1,a

AC2,a

A Cn,a

A T, 32d

and the matrix representations of the Greens functions be-come

GR = a

CaR

CaA

E a=

CR

CA

E +

CRC

A

E + i/2,

33a

GA = a

CaACa

R

E a* =

CAC

R

E * +

CAC

R

E i/2,

33b

= + 0, 33c

where the subscript denotes molecular orbital energieswith an imaginary part and the subscript , and C

R and CA

denote those without an imaginary part. We remark that isinfinitely small, and the imaginary part of the last terms inEqs. 33a and 33b can be replaced with a delta functionsee Eq. 24. By substituting these relations in Eq. 14, thelesser Greens function becomes

G = i fE

CAC

R

E

* +

CAC

R

E

i

/2

CRC

A

E +

CRC

A

E

+ i

/2

= i fE

,

CAC

R

E *

CRC

A

E + i fE

,

CA

CR

E *

CRC

A

E + i/2+ i fE

,

CAC

R

E i/2

CRC

A

E

+ i fE ,

CAC

R

E i/2

CRC

A

E + i/2. 34

The molecular orbital energy without imaginary part means that the molecular orbital has no interactions with the semi-infinite electrodes, then C

R=CA see Appendix C. By noting that C

R=CA=C and Ca

RSCb

A =ab, the second term in Eq.34 becomes zero as follows:

second term = i fE ,C

ACR

E * iHmol + R

Hmol A

CRC

A

E + i/2

= i fE ,

iCA

CRHmol +

RCRC

A CA

CRHmol +

ACRC

A

E * E + i/2

= i fE ,

iCAC

RSCAC

R iCAC

RSCAC

R

E *E + i/2

= 0. 35

We can argue similarly for the third term in Eq. 34 andobtain the following equations as the lesser Greens functionand the density matrix:

G = i fE,

CA

CR

E *

CR

CA

E

+ i fE

CC

E i/2

CC

E + i/2, 36

D = 12

+fE

,

CAC

R

E *

CRC

A

E dE

+

+

fE

CC 1

2

E 2 + /2

2 dE

=1

2

+

fE,

CA

CR

E *

CR

CA

E dE

+

+

occ

CCE dE. 37




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Equation 24 is used to change the first line to the sec-ond line in Eq. 37. In our calculations Eq. 37 is used as

the NEGF-based density matrix.

III. THE NEGF-BASED HARTREE-FOCK APPROACH

In this paper, we calculate the Fock matrix by using theHartree-Fock method and the density matrix based on Eq.37. In these calculations, we determine the Fock matrix inthe external electric field for electrodes with different appliedbiases. The NEGF-based Fock matrix Hmol can be deter-mined from the following equations by using the NEGF-

based density matrix of Eq. 37:

Hmol= h+

Dg g ,

38a

h= h = r 122rdr

+ A

r ZARA rrdr F rrrdr,

38b

g = r1r2 1r1 r2r1r2dr1dr2,38c

where h includes the kinetic energy, the nucleus-electroninteraction, and the interaction with the external electric

field. g is an electron-electron Coulomb repulsion operator, Fis the external electric field, and ZA is the atomic number ofnucleus A.

In our calculations, we perform iteratively the followingsteps until the NEGF-based density matrix converges, asseen in Fig. 3. The Fock matrix, which includes the interac-tion with the external electric field, is calculated based on theNEGF-based density matrix and Eq. 38a, and then the ex-trapolated Fock matrix is obtained by the direct inversion inthe iterative subspace DIIS method.48 Next, the eigenvec-tors and eigenvalues are calculated from the extrapolatedFock matrix. Then, the NEGF-based density matrix is recal-culated from the extrapolated Fock matrix and the self-energy; the eigenvectors and eigenvalues have to be deter-

TABLE I. Mulliken charges on C6H4S2.

Bias V 0.0 1.0 2.0 3.0

Without NEGFS1 0.0572 0.0287 0.0003 0.0279C1 0.0685 0.0645 0.0606 0.0568C2 C6 0.0740 0.0781 0.0823 0.0863C3 C5 0.0740 0.0698 0.0657 0.0616C4 0.0685 0.0726 0.0768 0.0810H1 H4 0.0797 0.0764 0.0732 0.0699H2 H3 0.0797 0.0829 0.0861 0.0894S2 0.0572 0.0857 0.1143 0.1429Total 0.0000 0.0000 0.0000 0.0000

NEGFS1 0.0873 0.0666 0.0606 0.0652C1 0.0655 0.0616 0.0570 0.0531C2 C6 0.0700 0.0730 0.0739 0.0732C3 C5 0.0700 0.0652 0.0587 0.0519C4 0.0655 0.0689 0.0708 0.0716H1 H4 0.0846 0.0824 0.0826 0.0847H2 H3 0.0846 0.0888 0.0952 0.1034S2 0.0873 0.1227 0.1727 0.2316Total 0.1016 0.1250 0.1960 0.2981

FIG. 3. SCF loop in the NEGF-based HF approach.

FIG. 4. Coordinates in our calculations.

FIG. 5. Change of total charges on C6H4S2.




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mined at each energy of the electron wave because the self-energy term depends on its energy. Here, we note that thesecond term in Eq. 37 is, approximated by using the eigen-vectors of the extrapolated Fock matrix. If the calculatedNEGF-based density matrix does not converge, we return tothe first procedure again and continue the above steps untilthe density matrix converges.

The NEGF-based Mulliken population and the chargeQA on nucleus A can be calculated by the following equa-tions:

A = A

AO

AO

DS, 39a

QA = ZA A, 39b

where the density matrix D is calculated based on Eq. 37.

IV. CALCULATIONS FOR BENZENEDITHIOL

In this section, we discuss the calculated result of elec-

tron transfer through benzenedithiol by using the NEGF-based HF approach. In our calculations, diffuse functions areadded to the STO-6G basis in order to let the molecule re-spond to the external electric field. The parameters for thesurface Greens function in Eq. 3b are set to be =4.77 eV, =3.0 eV, and c=3.5334 . We also assumethat the electrodes can interact only with the 3py orbital pitype of the sulfur atom of the C6H4S2 species. Here, the z

axis goes through the S1 and S2 sulfur atoms of C6H4S2, andthe x axis is perpendicular to the z axis and in the plane ofC6H4S2 see Fig. 4. Although highest occupied molecularorbital HOMO, HOMO-4, and lowest unoccupied molecu-lar orbital LUMO can interact with the electrodes underthis assumption, the HOMO is expected to dominate the total

current through benzenedithiol because it is closest to theFermi level.We calculated both the ordinary free molecule and

NEGF-based Mulliken charges see Fig. 5. Table I showsthat the total sum of the NEGF-based Mulliken charges is notzero, unlike ordinary Mulliken charges, because of electrontransfer from C6H4S2 to the electrodes. In addition, theNEGF-based Mulliken charges change according to the ap-plied voltage on the electrodes; Fig. 5 shows the change oftotal charge on C6H4S2. The chemical potentials p and q,respectively, for leads p and q, are summarized in Table IIwith the HOMO energy HOMO

0 and the first energy correc-tions Re

HOMO

1 and Im HOMO

1 calculated by using simulta-neous equations.20,21 Figure 6 shows the relationship be-tween these energies, which are fitted to a Lorentz functionin order to explain the transmission probability. When thebias increases, lead p extracts the electron from C6H4S2 as isclearly seen in the relationship between these energies. Onthe other hand, the contribution of electron occupation fromlead q induces only a small change in the total electron oc-cupation because the electron from lead q almost fills theHOMO even in the case of a low bias, and more positivecharges resulting on C6H4S2 with a high bias. Table II also

TABLE II. Relationship between HOMO energy and chemical potentials ofelectrodes.

Bias V 0.0 1.0 2.0 3.0

HOMO0 5.79 5.86 6.07 6.36

Re HOMO1 a 0.03 0.03 0.04 0.06

Im HOMO1 a 0.201 0.201 0.200 0.198

HOMO0 +Re HOMO

1 a 5.82 5.89 6.11 6.42p

a 4.77 5.27 5.77 6.27q

a 4.77 4.27 3.77 3.27Emolecule

b 1021.413 1021.409 1021.394 1021.373

aIn eV.bIn a.u.; defined in Eq. 40.

FIG. 6. Relationship between HOMO energy and chemical potentials of theelectrodes.

FIG. 7. Change of energy of molecule according to the applied bias.

FIG. 8. Change of HOMO energy according to the applied bias.




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shows that the HOMO energy shifts in the negative directionwith an increasing bias. In order to investigate why the mo-lecular orbital shifts in the negative direction, we analyzedthe NEGF-based energy for C6H4S2 by the followingequation:

Emolecule = ,

n

Dh+1

2 ,,,

n

DDg

g + Vnn, 40

where D and D are the NEGF-based density matrices inEq. 37, and Vnn is the repulsion energy between nuclei. Theresults are summarized in Table II and the change in Emoleculeis shown in Fig. 7 with the energy of the molecule withequilibrium Greens function EGF for comparison. The in-crease in the NEGF-based energy indicates that the electrontransfer to the electrodes destabilizes C6H4S2, so the mol-ecule responds to prevent transport of an electron from itselfto the electrode by lowering the HOMO energy. The chemi-cal potential of lead p and the HOMO energy with respect tothe applied bias are shown in Fig. 8. The HOMO energy

lowers with the decrease of the chemical potential of lead p,although the HOMO energy with EGF is almost constant.Therefore, there are two reasons why the molecular orbitalshifts: one is the effect of self-energy representing semi-infinite electrodes, and the other is the charge transfer from amolecule to the electrode.

The computed electrostatic potentials, expressed as en-ergies, calculated from NEGF-based Mulliken charges and

the three-dimensional Poisson equation are shown in Figs.9c and 9d; Figs. 9a and 9b are for ordinary Mullikencharges with EGF. The NEGF-based potential energies onthe sulfur atoms in C6H4S2 are particularly different fromthose with EGF due to electron transfer from molecule toelectrodes. Lastly, theoretical currents through C6H4S2 basedon Landauers formula and its conductance are respectively,given in Figs. 10a and 10b; the currents and conductancein Fig. 10a are determined with EGF, and those in Fig.10b are determined by the NEGF-based HF approach.From these figures, we can easily overview the influence ofelectron transfer from the molecule to the electrodes.

FIG. 9. Potential energies of C6H4S2 at a bias voltage of 3.0 V: a by EGF for y =0.0 surface, b by EGF for y =1.0 surface, c by NEGF for y=0.0 surface, and d by NEGF for y =1.0 surface.




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V. SUMMARY

In this paper, we have discussed a NEGF-based HF ap-proach. Although the NEGF method has so far been mainlyadopted with DFT, the clarity of the HF approach can easilygive a deep understanding of molecular conduction. In bothDFT and HF, the NEGF-based density matrix can reduce to

an ordinary density matrix if there are no interactions be-tween a molecule and an electrode, and NEGF-based Mul-liken charges have similar properties to those of free mol-ecules. Moreover, the only difference in the self-consistentfield SCF cycles between the ordinary and NEGF-basedHF approaches is a routine to calculate the density matrices.That is, the NEGF-based HF approach is the same as theordinary approach if the molecule does not interact with theelectrodes. These characteristics of the NEGF-based HF ap-proach are useful in the study of complicated phenomena inmolecular conduction because we can directly compare mol-ecules connected to electrodes with isolated ones. On theother hand, although the electron-electron correlations are

considered in the DFT method, the choice of functional re-mains arbitrary. If NEGF-based post-HF methods, such asconfiguration interaction CI methods40 and Mller-Plessetmethods, are adopted, we will be able to more easily inves-tigate the effects of electron-electron correlations and willgain deeper insights into electron-electron correlation in mo-lecular conduction. The NEGF-based HF approach is a basisof these post-HF methods for molecular conduction.

We also calculated the transmission probability for trans-port junction using C6H4S2 based on the NEGF-based HFapproach. In our calculations, we paid attention especially toelectron transfer through the HOMO because it dominatesfor C6H4S2, at biases below 2 eV. We used a highly over-

simplified tight-binding model for the electrodes. Calcula-tions by the NEGF approach gave electron transfer from themolecule to the electrodes and shifted the molecular orbitalenergy. The electron transfer to the electrodes destabilizesC6H4S2, and the HOMO shifts in a direction so as not toincrease the electron transfer. The calculations for otherproperties by NEGF-based HF approach are in progress.Those results will be reported elsewhere.

ACKNOWLEDGMENTS

This research was supported by a Grant-in-Aid for The21st Century COE Program for Frontiers in FundamentalChemistry, and for Scientific Research KAKENHI in Pri-ority Area Molecular Nano Dynamics, from the Ministryof Education Science Sports and Culture of Japan. The au-thors thank the Computer Center of the Institute for Molecu-lar Science for the use of computers. Two of the authors M.R. and Y. X. thank the MURI/DURINT program of the NSFfor support.

APPENDIX A: DERIVATION OF LANDAUER FORMULA

In this appendix, we derive the form of the current in Eq.6a from the Greens functions.16,34,44 The current operatorcan be expressed as follows:

jt = ed

dtne

= ei

H,ne =

e

hHG GH

= e

hHmolG

+ G GHmol G, A1

where H=Hmol+ and the charge density of electron ne isexpressed by the lesser Greens function see Eq. 10a. Weconvert the current operator from the time domain to thefrequency domain energy domain by Fourier transforma-tion. The lesser Greens function can be expressed by usingthe self-energy as follows:

G = GRGA. A2

Substituting Eq. A2 in Eq. A1 gives

jE = e

hHmolG

RGA + G GRGAHmol G .

A3

We obtain the following equations from Eq. 4b:

HmolGR = ESGR RGR I, A4a

GAHmol = GASE GAA I, A4b

and substitution of the above equations in Eq. A3 gives

FIG. 10. Theoretical currents through benzenedithiol and conductance aby EGF- and b NEGF-based HF approaches.




10/12

jE = e

hESGRGA RGRGA GA + G

GRGASE+ GRGAA + GR G

= e

hESG RG GA + G GSE

+ GA + GR G. A5

We can calculate the current from a trace operation,

jE

= TrjE = e

hTr R AG + GR GA

= ie

hTrG A , A6

where we use Eqs. 12a and 12b and the relation TrAB=TrBA. If=p

+q and =p +q, Eq. A6 becomes

jE = jpE + jqE, A7a

where

jpE = ie

hTrpG

p

A, A7b

jqE = ie

hTrqG

q

A. A7c

Noting from Eqs. 13 and A2

= if, A8

we obtain

jpE = i

e

h Tr

pi fpGR

pG

A

+ i fqGR

qGA

i fppGRp + qG

A =e

hTrpG

RqGAfq

fp. A9

APPENDIX B: DERIVATION OF APPROXIMATEDENSITY MATRIX

We derive the equation for the density matrix when weuse approximate Greens functions. This method is useful ifit is necessary to accommodate large systems, such as DNAor proteins.

We obtain the following equation from Eq. 23:

Im a1 =

1

2,C,a

0C,a

0 , B1

and another expression for Eq. 21,

Im GR =

a

C,a0 C

,a0 12, C,a0C,a0

E a0 Re a

12 + 12, C,a0C,a02

= a

1

2, C,a0

C,a0C,a

0

C,a0

E a0 Re a1 + i2, C,a0C,a0E a0 Re a1 i

2, C,a0C,a

0

= 12 a

,

C,a0 C,a

0

E a0 Re a

1 + i Im a1

C,a0 C

,a0

E a0 Re a

1 i Im a1 =

12

a

gaAga

R, B2

where

gaR=

C,a0 C

,a0

E a0 Re a

1 i Im a1 , B3a

gaA=

C,a0

C,a0

E a0 Re a

1 + i Im a1 . B3b

Then, the approximate density matrix is given as

D = 1

Im GR =

1

2aga

Aga

R. B4

APPENDIX C: RELATION IN MOLECULAR ORBITALS

We derive the relation CR

= CA

in this appendix. Weconsider the case of Im =0,

Hmol + RC

R = SCR. C1

The self-energy R can be obtained from Eq. 1, and weconsider one-dimensional electrodes in order to simplify thederivations. The self-energy can be divided into two parts:

R = 1R + 2

R = a

a=,

aT 1

aexpikacaa, C2a




11/12

1R

T 1

coskc, sinkc = 0, C2b

2R

T 1

coskc + i sinkc,

sinkc 0, C2c

where a =t1,at2,a

tn,a. Then, the first-order correction oforbital energies can be obtained as

Hmol + 1RC

0

= a0SC

0, C3a

a1 = C

0T2RC

0

= C0T Re 2

RC0 + iC

0T Im 2RC

0. C3b

With the case of Eqs. C2b and C2c,

1 becomes asfollows:

1

= C0

T

2

R

C0

=

1

coskc

r

n

Cr,0tr,

2

+ i

1

sinkc

r

n

Cr,0tr,

2

, C4

where

C0T= C1,

0C2,0 Cn,

0 C5

Therefore, the following relationship can be derived in the

case of Im R=0:

r

n

Cr,0tr,

2

= 0, C6

and

2R

C0 =

a

r,s

n

tr,ats,a1

aexpikacaCs,

0

= a

r

n

tr,a1

aexpikaca

s

n

ts,aCs,0 = 0.

C7

Finally,Hmol +

RC0 = Hmol + 1

RC0 =

0SC0, C8

Hmol + AC

0 = Hmol + 1AC

0 = 0SC

0. C9

Therefore, CR= C

A= C

0 by noting 1R =1

A.In multidimensional cases, the surface Greens function

can be obtained as follows:

gsurfaceR =

a

uauaT expikaca, C10

where ua is the real column vector, and 1 is

1 = C

0T2RC

0

=

C0TTuu

TC0 expikc

=

uTC0TuTC

0 expikc . C11

Equation C11 has the similar formula with Eq. C4, there-fore, we can derive the same relations as Eqs. C8 and C9.

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