Introduction to the tight bindingapproximation—implementation by

diagonalisation

Anthony T Paxton

Atomistic Simulation CentreSchool of Mathematics and Physics

Queen’s University Belfast

http://titus.phy.qub.ac.uk/group/Tony/WSMS2009

WSMS–Julich, March 4, 2009

Outline

1. Tight binding approximations2. Bond integrals and Slater Koster table3. Energy bands and DOS in the simple cubic structure4. Self consistent tight binding5. Tight binding in small molecules—some new results

1 / 27

WSMS–Julich, March 4, 2009 Energy bands in Ge

pseudopotential tight binding free electron

One electron dispersion relations in germanium (after Harrison)

2 / 27

WSMS–Julich, March 4, 2009 Tight binding approximation—I

The Bloch sum of s–orbitals at N sites {R}

ψk(r) =1√N

XR

eik·R

ϕs (r−R)

obeys Bloch’s theorem because if R′ is another atomic site

ψk(r + R′) =

1√N

eik·R′ X

R′′=R−R′eik·R′′

ϕs`r−R

′′´= e

ik·R′ψk(r)

If ψk is an eigenstate of the Schrodinger equation H0ψk = εkψk then its eigenvalue willbe

εk =

Rdr ψkH

0 ψkRdr ψk ψk

3 / 27

WSMS–Julich, March 4, 2009 Tight binding approximation—II

H0

= −~2

2m∇2

+ Veff (r) Veff (r) =XR′′

v(r−R′′

)

In which case we have

ZψkH

0ψk =

1

NXR,R′

eik·(R−R′)

Zϕs`r−R

′´24− ~2

2m∇2

+XR′′

v(r−R′′

)

35ϕs (r−R)

Zψk ψk =

1

NXR,R′

eik·(R−R′)

Zϕs`r−R

′´ϕs (r−R)

The tight binding approximations

1. Neglect three centre integrals—i.e., those for which R 6= R′ 6= R′′

2. Neglect overlap integrals (except those for which R = R′)

3. Don’t even attempt to calculate the remaining integrals—treat them as disposableparameters of the TB model, retaining only nearest neighbours.

4 / 27

WSMS–Julich, March 4, 2009 Tight binding approximation—III

Then we find

εk = εs +

Zϕs(r)

24XR6=0

v (r−R)

35ϕs(r) +XR6=0

eik·R

Zϕs(r)v(r)ϕs(r−R)

εs =

Zϕs(r)

"−

~2

2m∇2

+ v(r)

#ϕs(r)

If we neglect the crystal field terms,

εk = εs +XR6=0

eik·R

h(R)

+ +

Rh(R) = h = ssσ

So, given some hopping (or transfer) integrals and a crystal structure we can construct the

energy bands.

5 / 27

WSMS–Julich, March 4, 2009 Bond integrals

+ +

+ +

+

+

+

+

+

+++

+ +

+

+

+

+

+

+

+

−

−

−

−−−

−−

−

−−

−−

−

−−

+ + −+

+

+

+

− −

−

−

− −ppπ

ddσddπ

ssσ spσ

ppσ

sdσ

pdπpdσ

ddδ

6 / 27

WSMS–Julich, March 4, 2009 The simple cubic s–band

For example in the simple cubic s–band with lattice constant a

{R} = { [±a, 0, 0] [0,±a, 0] [0, 0,±a] }

εk = εs +XR6=0

eik·R

h(R)

= εs + 2ssσ (cos kxa+ cos kya+ cos kza)

Plot this along high symmetry lines in the Brillouin zone, Γ: (000), R: πa (111), X: πa (100)

−6

0

6

−(ε−

ε s)

/ ssσ

R Γ Xn(ε)

7 / 27

WSMS–Julich, March 4, 2009 p–orbitals—I

The three p-orbitals are ϕx(r−R), ϕy(r−R) and ϕz(r−R). The expansion of a trialwavefunction or Bloch sum is

ψk(r) =1

NX

α=x,y,z

cαXR

eik·R

ϕα(r−R)

The Schrodinger equation becomes a secular problem˛(εp − εk) δαα′ + Tαα′

˛= 0

Tαα′ =XR6=0

eik·R

Zϕα(r)v(r)ϕα′ (r−R)

8 / 27

WSMS–Julich, March 4, 2009 p–orbitals—II

Zϕα(r)v(r)ϕα′ (r−R) = h(R)

+

θ

R

+−

−

h(R) = ppσ cos2 θ + ppπ sin2 θ

generally, if l, m and n are the direction cosines of RZϕxvϕx = l

2ppσ +

“1− l2

”ppπZ

ϕxvϕy = lm ppσ − lm ppπZϕxvϕz = ln ppσ − ln ppπ

Bond Integrals 9 / 27

WSMS–Julich, March 4, 2009 The Slater–Koster table, s–p block

10 / 27

WSMS–Julich, March 4, 2009 The Slater–Koster table, sp–d block

11 / 27

WSMS–Julich, March 4, 2009 The Slater–Koster table, d–d block

12 / 27

WSMS–Julich, March 4, 2009 The simple cubic p–band—I

You don’t even need to diagonalise the secular matrix in the simple cubic crystal structuresince it’s already diagonal!!

Tαα′ = Tααδαα′

and

Txx = 2 ppσ cos kxa+ 2 ppπ (cos kya+ cos kza)

Tyy = 2 ppσ cos kya+ 2 ppπ (cos kza+ cos kxa)

Txx = 2 ppσ cos kza+ 2 ppπ (cos kxa+ cos kya)

In the (100) direction, Γ→ X, k = (kx00) there are three bands,

εk = εp +

8><>:2 ppσ cos kxa+ 4 ppπ

2 ppπ + 2 (ppσ + ppπ)

2 ppπ + 2 (ppσ + ppπ)

≈ εp +

8><>:2 ppσ cos kxa

2 ppσ

2 ppσ

13 / 27

WSMS–Julich, March 4, 2009 The simple cubic p–band—II

−2

0

2

(ε−ε

p) /

ppσ

R Γ Xn(ε)

Γ→ X : εk ≈ εp +

8><>:2 ppσ cos kxa

2 ppσ

2 ppσ

14 / 27

WSMS–Julich, March 4, 2009 Bands of Strontium Oxide

−0.4

−0.2

0

0.2

0.4E

nerg

y (R

y)

SrO lmf

Γ M X R Γ XM R−0.2

0

0.2

0.4

0.6

0.8

Ene

rgy

(Ry)

SrO TB

Γ M X R Γ XM R

Γ→ X : εk ≈ εp +

8><>:2 ppσ cos kxa

2 ppσ

2 ppσ

15 / 27

WSMS–Julich, March 4, 2009 Further. . .

• d–bands

• hybridisation

• non orthogonal tight binding

• total energy and force

• self consistent tight binding

• magnetic tight binding

• small molecules

• time dependent tight binding

”Fools rush in where angels fear to tread,” Alexander Pope 1711

16 / 27

WSMS–Julich, March 4, 2009 self consistent tight binding

There are components of the electrostatic potential at site R due to all multipoles at sites R′:

VRL = e2XR′L′

BRLR′L′ QR′L′

B is a sort of generalised Madelung matrix, proportional to˛R−R′

˛−1 for point charges.They induce electrostatic shifts in the hamiltonian matrix elements which are on site and offdiagonal:

H′RL′ RL′′ = UR ∆qR δL′L′′ +

XL

VRL ∆`′`′′` CL′L′′L

The Hubbard–U acts to resist charge accumulation.

H = H0

+H′

Crystal field 17 / 27

WSMS–Julich, March 4, 2009 Zirconia bands

Rutile (LDA) Rutile (TB)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Ene

rgy

(Ry)

Γ Z R A M Γ Z0.2

0.4

0.6

0.8

1

1.2

1.4

Ene

rgy

(Ry)

Γ Z R A M Γ X

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Ene

rgy

(Ry)

XUK Γ X W L Γ0.2

0.4

0.6

0.8

1

1.2

1.4

Ene

rgy

(Ry)

XUK Γ X W L Γ

Fluorite (TB)

t2

t2

e

e

Fluorite (LDA)

18 / 27

WSMS–Julich, March 4, 2009 Zirconia phase transition

S. Fabris, A. T. Paxton and M. W. Finnis,PRB, 63 094101 (2001)

57 H. M. Kandil, J. D. Greimer, and J. F. Smith, J. Am.Ceram. Soc. 67, 341 (1982).FIG. 1. Cubic and tetragonal structures of ZrO2. Lightand dark circles denote oxygen and zirconium atoms respec-tively. Arrows represent the structural instability of the oxy-gen sublattice along the X�2 mode of vibration.FIG. 2. Time dependent order parameters at 700 K: �x and�y oscillate around 0, �z around ��z, the value of the macro-scopic order parameter.

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5

δ (a

.u.)

t (ps)FIG. 3. Time correlation functions (top) and correspon-dent Fourier transforms (bottom) of the time dependent orderparameters �x, �y, and �z.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

S(t)

t (fs)

zx,y

05 10 15 20 25

S(ν)

ν (THz)

ν z-

zx,yFIG. 4. Temperature dependence of the macroscopic orderparameter ��z. The symbols (�) are the results of the calcu-lations. The continuous solid eye-line is extrapolated in theregion near Tc where the large uctuations in �z make theaveraging procedure inaccurate.

0

0.1

0.2

0.3

0.4

0.5

0.6

600 800 1000 1200 1400 1600 1800 2000 2200

<δ>

(a.

u.)

T (K)FIG. 5. Calculation results of the frequency squared ��2z vs.temperature for two simulation sets : (a) tetragonal cell withtemperature dependent lattice parameters taken from exper-iment,6 (b) Cubic cell with temperature independent latticeparameter (see text).0

10

20

30

40

50

600 800 1000 1200 1400 1600 1800 2000 2200

ν z--2 (

TH

z2 )

T (K)

(a)

(b)FIG. 6. Section of the 0 K energy surface for the cubic cell.The isoenergetic contours are plotted on the base every 0.3mRy/ZrO2.12

-0.4 -0.2 0 0.2 0.4δx (a.u.)-0.4

-0.20

0.20.4

δy (a.u.)

-2-10

∆U (mRy)FIG. 7. 0 K energy surfaces for a tetragonal cell with thetetragonal axis along z: (a) section in the �y; �x plane, (b) sec-tion in the �z; �x plane. The isoenergetic contours are plottedon the base every 0.3 mRy/ZrO2 .(a) (b)-0.4

-0.20

0.20.4

δy-0.4

-0.20

0.20.4

δx

-2-101

∆U (mRy)

-0.4-0.2

00.2

0.4

δz-0.4

-0.20

0.20.4

δx

-2-101

∆U (mRy)

1

FIG. 8. Transferability of the 0 K energy-expansion co-e�cients between di�erent tetragonal cells with the tetrag-onal axis along z. Projections of the corresponding energysurfaces along the high-symmetry order-parameter directionsh00�i (a), and h��0i (b).-3

-2

-1

0

1

0 0.1 0.2 0.3 0.4 0.5

∆U (

mR

y/Z

rO2)

<0,0,δ> (a.u.)

(a)

c/a=1.00c/a=1.01c/a=1.02

0 0.1 0.2 0.3 0.4-3

-2

-1

0

1

<δ,δ,0> (a.u.)

(b)

c/a=1.00c/a=1.01c/a=1.02FIG. 9. Temperature dependent free energy gradients cal-culated using Eq.(7) and corresponding �t via the analyticform derived from the Landau theory (1). Projection alongthe order parameter direction h00�i.

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.1 0.2 0.3 0.4 0.5 0.6

dF/d

δ (a

.u./Z

rO2)

<0,0,δ> (a.u.)

T=2000 K

T=50 K

T=50 KT=500 KT=1000 KT=1500 KT=2000 K

FIG. 10. Temperature evolution of the free energy pro�lesprojected along the order parameter direction h00�i. Thesymbols (�) are the 0 K calculations, the thick solid lines arethe result of the thermodynamic integration and correspondto the temperatures 50, 500, 1000, 1500, and 2000 K.-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

∆F (

mR

y/Z

rO2)

<0,0,δ> (a.u.)

T=0 K

T=2000 K

T

FIG. 11. Double well in the internal energy along the orderparameter direction h00�i: the solid line correspond to the 0K calculation, the symbols are the averaged internal energiesfrom the constrained MD simulation.-2.5

-2

-1.5

-1

-0.5

0

0.5

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

∆U (

mR

y/Z

rO2)

<0,0,δ> (a.u.)

T=0 KT=50 KT=500 KT=1000 KT=2000 KFIG. 12. Entropic contribution to the phase transition ob-tained by applying the de�nition of the Helmholtz free energy�F = �U � T�S to the data shown in Figures 10 and 11.13

19 / 27

WSMS–Julich, March 4, 2009 TB model for water

Finding TB parameters for water

20 / 27

WSMS–Julich, March 4, 2009 Water monomer

−2q

θ

p(δ)

q = δeq

pind

d

point charge model dipole polarisable model

p(δ) = 2dδe cos 12 θ = 5.65δ (Debye) ; e > 0

pind = −αO(δ)2eδ

d2cos 1

2 θ ;pind

p(δ)= −

αO(δ)

d3

pexp = p(δ) + pind = 1.86 (D)

αO(δ) ≈ 0.8− 0.23δ → δ ≈ 0.9 ; (N→ O→ F→ Ne)

δ αO ν1 ν2 ν3 αH2O Ecoh

model (A3

) —force constants (au)— (A3

) (Ry)

point 0.33 — 1.029 0.099 1.002 1.3 0.79dipole 0.45 0.31 1.029 0.104 0.847 1.2 0.84exp. — 0.7 1.029 0.100 1.062 1.4 0.75

21 / 27

WSMS–Julich, March 4, 2009 Water Dimer

non self consistent model point charge model dipole polarisable model

R rf rd α βTB 291 95.4 97.3 8.8◦ 97.6◦

CCSD 291 95.7 96.4 5.5◦ 124.4◦

Computational determination of equilibrium geometry and

dissociation energy of the water dimer¤

W. Klopper, J. G. C. M. van Duijneveldt-van de Rijdt and F. B. van Duijneveldt

T heoretical Chemistry Group, Debye Institute, Utrecht University, P.O. Box 80052, NL -3508 T B

Utrecht, T he Netherlands

Received 23rd December 1999, Accepted 11th February 2000

Published on the Web 6th April 2000

The equilibrium geometry and dissociation energy of the water dimer have been determined as accurately as

technically possible. Various quantum chemical methods and high-quality basis sets have been appliedÈthat

is, at the level of a nearly complete basisÈand both the intermolecular separation and the deformation of the

donor and acceptor molecules have been optimized at the level of CCSD(T) theory (coupled-cluster theory

with singles and doubles excitations plus a perturbation correction for connected triples). It is found at the

CCSD(T) level that the monomer deformation in the dimer amounts to 86% of the deformation computed at

the MP2 level (second-order perturbation theory) and that the core/valence electron correlationMÔller-Plesset

e†ects at the CCSD(T) level amount to 80% of the same e†ects at the MP2 level. The equilibrium OÉ É ÉO

distance is determined as pm and the equilibrium dissociation energy as kJRe\ 291.2 ^ 0.5 D

e\ 21.0 ^ 0.2

mol~1, with respect to dissociation into two isolated water molecules at equilibrium. Accounting for zero-point

vibrational energy, the theoretical prediction for the dissociation energy becomes kJ mol~1, aD0

\ 13.8 ^ 0.4

result which is open to direct experimental veriÐcation.

1 Introduction

Rapid progress has recently been made in the precise determi-nation of the equilibrium geometry and binding energy of thewater dimer by ab initio methods.1h33 Apart from advances inhardware and software the progress has been stimulated bythe availability of basis sets which (in principle) allow a sys-tematic approach to the basis set limit at a given level oftheory, and by a growing consensus that the basis set super-position error (BSSE) can rigorously be avoided by applyingthe counterpoise method (CP), so that the user can focus on agood description of the interaction without having to worryabout the size of the BSSE. A survey of recent studies, givingdetails of the geometries (cf. Fig. 1) obtained or explored, isgiven in Table 1.

A fairly complete set of results converging to the basis setlimit is now available at the MP2/CP level of theory.Xantheas21 has converged the OÉ É ÉO equilibrium distance R

e

Fig. 1 The equilibrium structure of the water dimer.

¤ Dedicated to Professor Reinhart Ahlrichs on the occasion of his60th birthday.

to shorter and shorter values, reaching pm for theRe\ 291.3

aug-cc-pV5Z basis. His results agree with those of fullgeometry optimizations using the counterpoise procedurereported by Hobza et al.27 In rigid monomer calculationsusing conventional basis sets the frozen-core interactionenergy, *E(FC), approaches the limit from above to a valuearound [20.5 kJ mol~1, the lowest reported values being[20.28 kJ mol~1 (ref. 25) and [20.36 kJ mol~1 (refs. 24 and32). Corresponding MP2-R12 calculations, which may betaken to represent the basis set limit, give [20.51 kJ mol~1.31Relaxing the monomer geometries lowers *E by about [0.2kJ mol~1 (refs. 21 and 27) and the introduction of coreÈvalence correlation e†ects lowers *E by a further [0.2 kJmol~1.24,31,32 Thus the Ðnal MP2 interaction energy takingall these contributions into account will be about [20.9 kJmol~1.

At higher levels of theory the picture is less complete.Halkier et al.25,33 have shown that at their chosen dimergeometry, the step from MP2 to CCSD(T) changes the rigidmonomer *E(FC) by amounts varying from ]0.13 kJ mol~1for the aug-cc-pVDZ basis to [0.24 kJ mol~1 for aug-cc-pVQZ, and their extrapolations to the basis set limit suggestan increase to [0.27 kJ mol~1. Klopper and have esti-Lu� thimated the increase to [0.2 ^ 0.1 kJ mol~1.31 As for themonomer relaxation e†ects, it is known that the OHlengthening is exaggerated at the MP2 level,14,20 so oneexpects a smaller relaxation e†ect for CCSD(T). The size ofthe coreÈvalence correction at the CCSD(T) level has not yetbeen studied.

The aim of the present paper is to study these aspects at theCCSD(T) level of theory in an e†ort to obtain as accurate aresult as is technically possible. At Ðrst sight, it does not seemcurrently feasible to generate CCSD(T) data which are asaccurate as those available for MP2. For example, to carryout CCSD(T) geometry optimizations in basis sets such asaug-cc-pV5Z, with 572 functions, is out of the question. Our

DOI: 10.1039/a910312k Phys. Chem. Chem. Phys., 2000, 2, 2227È2234 2227

This journal is The Owner Societies 2000(

W. Klopper et al., PCCP, 2, 2227 (2000)

22 / 27

WSMS–Julich, March 4, 2009 Water Tetramer

—dipole moment (D)—monomer 1.82dimer 1.97 1.98trimer 2.22 2.23 2.24tetramer 2.25 2.25 2.27 2.27

.

.

.bulk water ≈ 2.66 (point charge model)

23 / 27

WSMS–Julich, March 4, 2009 Polarisability of small molecules

−2 −1 0 1 2−3

−2

−1

0

1

2

3

electric field (GV/m)

dipo

le m

omen

t (D

ebye

)azulene

−4 −2 0 2 40

2

4

6

8

10

12

electric field (GV/m)

dipo

le m

omen

t (D

ebye

)

p-nitroaniline

N N

O

O

H

H

+−

NNO

O

H

H

++

−

−

24 / 27

WSMS–Julich, March 4, 2009 Time dependent TB

−0.01

0

0.01

0.02∆q

(el

ectr

ons)

0 500 1000 1500 2000 2500−30

−20

−10

0

10

20

30

time (fs)

π−b

ond

curr

ent (

µA)

770 780 790time (fs)

dρ

dt=

1

i~[H, ρ]− Γ (ρ− ρ0) ; jRR′ =

2e

~XLL′

HR′L′ RL Im ρRLR′L′

25 / 27

WSMS–Julich, March 4, 2009 Ehrenfest dynamics

−0.01

0

0.01

0.02∆q

(el

ectr

ons)

0 500 1000 1500 2000 2500−30

−20

−10

0

10

20

30

time (fs)

π−b

ond

curr

ent (

µA)

560 570 580time (fs)

MRd2R

dt2= −

XRLR′L′R′ 6=R

2ρRLR′L′ ∇RH0R′L′ RL −

XL

QRL∇RVRL −∇REpair

26 / 27

WSMS–Julich, March 4, 2009 Acknowledgements

Thanks to

Catherine WalshAlin Elena

Jorge KohanoffTchavdar Todorov

Mike Finnis

27 / 27