Lecture Krash 1

March 4, 2012 Semiempirical methods 1

CSC Spring School in Computational Chemistry

CSC Spring School in Computational Chemistry

SemiSemi--empirical and tightempirical and tight--binding methodsbinding methods

Arkady Krasheninnikov, University of Helsinki and

Aalto University


OutlineIntroductory remarks Quantum chemistry semi-empirical methodsSemi-empirical methods as a simplification of

the Hartree-Fock-Roothaan methodTight-binding methodThe Slater-Koster schemeExamples of tight-binding modelsThe power of simple tight-binding: predicting

the electronic structure and bonding trends in solids (an excursion to solid state physics research methods)


Semi-empirical methods: introductory remarks

Appeared in the 60th-70th as an attempt to find a compromise between the computational efficiency and accuracy

Let’s call the “real semi-empirical methods” (as opposed to the tight-binding methods) the methods which are close in spirit to the Hartree-Fock formalism but have fitted parameters, such as MINDO, MNDO, AM1, PM3, etc. (MOPAC)

Nowadays not that much in use (sometimes applied to big organic molecules and similar problems in organic chemistry)

The methods normally do not make use of the Slater-Koster approximation (to be discussed later on in this lecture); the one-electron integrals are assumed to be proportional to the overlap integrals which are evaluated through recursion formulas.

Semi-empirical methods are the methods which treat the electron system quantum mechanically (we solve the Schroedinger equation), but the true Hamiltonian is replaced with a model one; the parameters of the model Hamiltonian are fitted to reproduce the reference data (usually experiments)

The localized orbitals are used as basis functions

March 4, 2012 4

What you already know:Hartree-Fock-Roothan equations

Starting from the Fock equation and using a nonorthogonal basis set we can derive the following matrix equation (RHF):

wherek

k

k labels the orbitals

Sum over k runs over the occupied orbitals

Sums over p, q, r, s, run over the basis functions

March 4, 2012 5

Evaluation of the integrals

Two-electron integrals

We have to evaluate:

One-electron integrals:One-center integrals

Two-center integrals

One/two/three/four-center integrals As the number of basis functions is proportional to the number of atoms N, the time for integral evaluation scales as N4. Trick: drop out some integrals, as many integrals (three/four-center) are small: physically, negligible overlap of electron WF localized at different atoms : Still quite expensive… Further approximations/simplifications?

1 1 2 2


Approximations used in semi-empirical methods

We completely neglect all three- and four-center integrals

Only valance electrons are taken into account;

In many versions the overlap integrals in the secular (Hartree-Fock Roothaan) equation are ignored. Thus, we solve the ordinary eigen value problem instead of the generalized eigenvalue problem:

Practically all matrix elements are approximated by analytical functions of interatomic separation and atom environment. Parameters are chosen to reproduce the characteristics of reference systems.

Core-core (nucleus-nucleus) Coulombic repulsion is replaced with a parametized function.

Normally we still have a SCF loop, similar to the Hartree-Fock methods

Examples of the approximationsTo illustrate the typical approximations made, let’s recall the Hartree-Fock-Roothaan formulas:

where

In some of the semi-empirical methods the two-center one-electron integral is approximated by :

Two-electron two-center integral (for s orbitals centered at atoms A and B) is calculated as

Atom core repulsion

AMA, AMB, AB, AB compli-cated functions of atom types and separations

Or are calculated exactly through recursion formulas.

How well do these methods work?

Calculations of the cohesive (binding) energy for a carbon dimer by various semi-empirical methods incorporated into the MOPAC package

Which approximation should we trust? Note that we have up to 50 fitted parameters.

Alternative: tight-binding models which have much less parameters, physically are more transparent and possess more predictive power

Quality of a model: Quality = predictive power (transferability)

complexity


Introducing the tight-binding model

"Ockham's razor" principle:

It is vain to do with more what can be done with less.

OREntities should not be multiplied unnecessarily.

ORWith other things being equal, a simpler explanation is better than a more complex one

William of Ockham1285-1347

Maybe we could develop another theoretical model which would be:

As simple and having as few parameters as possible;

Physically well motivated;

Accurate enough to provide a qualitative (and even quantitative) description of the system.

As we will see, tight-binding has a very good transferability/complexity ratio!


Tight-binding models: general remarks

TB models became extremely popular in the 1980th, as they make simulations (geometry optimization and even molecular dynamics) for ~100 atoms tractable on the computers available at that time.

The method is frequently referred to as “Hückel model” in chemistry community

Besides this: If we fix the atomic structure, TB models (sometimes with one adjustable parameter) can be used for qualitative or even quantitative electronic structure calculations (even analytical).

Using special tricks like the recursion method and the moment’s theorem, one can calculate many informative characteristics (e.g., local density of states) without solving explicitly the Schrödinger equation. This makes it possible to calculate LDOS for a system composed of ~100000 atoms on a PC.

Simple TB models allow one to calculate very easily relative energies of different phases, that is to predict what phase is more stable.

The TB codes can be optimized relatively easily (linear-scaling algorithms).


Tight-binding models: general remarks (2)

TB models can conditionally be divided:

Orthogonal TB Non-orthogonal TB

Depending on how we treat the overlap matrix S:

Ab initio TB (derive from DFT)

Empirical (fit to experiments, or ab initio results)

Depending on how we derive the parameters:

Non-self-consistent Self-consistent

Depending on how we treat the charge self-consistency problem:

Important: in all modifications we can use the Hellmann-Feynman theorem


Deriving the tight-binding modelTight-binding method can rigorously be derived from density-functional theory:

Non-self-consistent TB: the first-order Hohenberg-Kohn-Sham functional)

Self-consistent TB: the second-order Hohenberg-Kohn-Sham functional)


Deriving the tight-binding model (2)Lets understand the main features of the TB model starting from the Hartree-Fockmethod.

The Hartree-Fock-Roothaan formulas:

where

Close in spirit to semi-empirical methods (TB is actually a semi-empirical method), let’s:

Neglect all two-electron integrals, unless all four basis functions are centered at the same atom (p=r=q=s):

For p=r=q=s we introduce a Hubbard-like correction to the total energy: q is the Mulliken charge on the atom p .

(not very common approximation, normally also omitted)


Deriving the tight-binding model (3)Further approximations:

One-electron integrals are not computed during the calculations; instead they are tabulated by analytical functions to be discussed later on;

The core repulsion energy (Long-ranged Coulombic repulsion is approximated by a short-ranged function:

The total electron energy is just a simple sum of electron orbital energies:

or

if we take into account the Hubbard correctionBy choosing the parameters and the shape of the functions we use instead of one/electron integrals we can match the reference data

occ

kCoulombk

HF VE 2


Slater-Koster approximationThe key moment is how we calculate one-electron integrals

Instead of introducing basis functions, e.g., Slater-like for the radial part and s,p,d for the angular part, then doing the integrals, we introduce an analytical function which depends on RAB only times another function which depends on the overlap of the angular functions localized at the atoms

Note that the matrix element depends on the positions of ALL atoms


Slater-Koster approximation (2)Let’s consider two atoms with s and p atomic orbitals (e.g. Carbon)

C

C

r

x

z

y

Atom 1:s px py pz

Atom 2:

Diagonal elements: Assume we know s and p energies (Es Ep ) for isolated single atom; we further assume that these “on-site” energies are the same when the bonds are formed

Off-diagonal elements:

s-s: Assume that for s-s atomic orbitals, wherer

x

x

is an analytical function which depends on r onlyis a parameter chosen to fit the results for a reference system


Slater-Koster approximation (3)Off-diagonal elements:

s-p: Assume that for s-p atomic orbitals,

is the same analytical function

is another parameter chosen to fit the results for a reference system

is the directional cosine of the vector between the atoms

xx

y

x

x r

+

x

x

y

yx

x

x

y x

-

+ +

+ - + +

+ -

+ +

l reflects the anisotropy o f the orbitals; note that p functions have a positive and negative part

Two-center one electron integrals are sometimes called hopping integrals


Slater-Koster approximation (4)Off-diagonal elements:

p-p: For px-px atomic orbitals, the matrix element is

Analogously, px-py matrix element is

pz-pz matrix element is

x

y

x

x

y

x

p-d: We can derive similar formulas for d orbitals as well

Four non-equivalent fundamental integrals between s and p atomic orbitals


Slater-Koster approximation (5)

Independent two center hopping integrals between atoms on the z-axis (from M. Finnis, “Interatomic forces in Condensed Matter)

So, all together, the integral look as follow:


Brief remark on the sign of some parameters

Quoting directly from M. Finnis, “Interatomic forces in Condensed Matter”, Oxford University Press, page 196:

Note that the sign of the matrix elements will be different:

sp ps

sp = -ps

“These sign changes are a fruitful source of errors for anyone writing their first tight-binding code (which is probably the best way to proceed if you really want to understand this stuff), simply because the matrix HIpIq (the Hamilton matrix) is only symmetric under the simultaneous interchange of I with J and p with q.”

At the same time, the Hamilton matrix must be symmetric:


Examples of TB modelsC.H. Xu, C.Z. Wang, C.T. Chang, and K.M. Ho, J. Phys. Condens. Matter 4 (1992) 6047).

Orthogonal tight-binding model for carbon:

Tight binding parameters:

Scaling function:

Additional trick: cutoff function at large separations, smoothly joint with s(r)Repulsion energy: r0, rc, d0, …

are parameters:

TBLDA


Examples of TB models (2)D. Porezag, et al., Phys. Rev. B 51, 12947 (1995)Non-orthogonal tight-binding model for carbon:

A Slater-type function was introduced first:

Solving the Kohn-Sham equation equations for these functions we derive the coefficients a in the expansion which were then used to calculate the TB parameters for the overlap and Hamiltonian matrices: Density-functional-theory tight binding (DFTB).The repulsion energy is fit to reproduce the results for the C-C dimer We solve the generalized eigenvalue problem and calculate the forces as:


Examples of TB models (3)

Review: T. Frauenheim et al. J. Phys.: Condens. Matter 14 (2002) 3015

Charge self-consistent non-orthogonal tight-binding model (extension of DFTB):

In the non-CSC formalism the total energy is:

With account for charge fluctuations:

The Hamiltonian matrix elements depend on the atomic charges as well:

Charge self-consistency is achieved through the iteration loop, similar to HFR:


DFTB: current state of the artSeminar on Monday 12 March 2012 at 14.15 in Lecture hall J (U149)

Thomas Frauenheim, Bremen Center for Computational Materials Science

DFTB+ - An approximate DFT method: Applications to computational nanomaterials.

The new release of DFTB+ as a density-functional (DFT)-based approach, combining DFT-accuracy and Tight-Binding (TB) efficiency, is reported; http//:www.dftb.org. Methodological details will be described briefly. Advanced functions include spin degrees of freedom, time dependent methods for excited state dynamics, efficient Order-N algorithms and multi-scale QM/MM/Continuum-techniques to treat reactive processes in nanostructures under environmental conditions. Additionally, the combination with non-equilibrium Green´s functions allows to address quantum transport in nanostructures and on the molecular scale.

Structure formation under technological relevant conditions is controlled by Molecular-Dynamics Simulated Annealing in ground and excited states. Successful applications are shown to cover a broad spectrum of problems, ranging from semiconductor nanostructures through amorphous materials to hybrid-interface design. Some detailed examples include studies of optical properties of functionalized semiconductor quantum dots and inelastic electron tunneling spectra in transport through single molecule junctions.Approaching new frontiers in materials science the DFTB-method is generalized and improved to cover also weak interactions and extended to QM/MM/Continuum-coupling for applications to functional biosystems. Recent work include the proton blocking in aquaporine channels and the spectral tuning in retinal protein photocycles. The QM/MM concepts, originally developed for biosystem-simulations are generalized to the study of bulk problems and surface adsorption

DFTB: available parameters

DFTB: available parameters (2)

DFTB: implementation


More about DFTBOverall, works well, providing a good compro-mise between accuracy and computational efficiency

Example: formation energies of single and double vacancies in carbon nanotubes as calculated by plane-wave DFT implemented in VASP (blue) and DFTB (red)

My personal opinion: DFTB (in general, TB) works better for insulators/semiconductors and monoatomic solids

Single (SV) anddouble (DV) vacancy formationenergy in carbonnanotubes

5 10 15diameter (Å)

2

3

4

5

6

7

ener

gy (e

V)

DV(a)

DV(z)

SV(a)SV(z)

PW DFT

PW DFT

Graphene edgestability

Another example

http://cst-www.nrl.navy.mil/bind/

A “universal” TB approach?

Computational efficiency and linear-scaling TB methods

Scaling of the simulation time in TB methods:

The most of time is spent for solving the eigen value problem. Note that the number of levels is comparable (normally a half) to the size of the Hilbert space (total number of basic functions). Thus, full diagonalization of the Hamilton matrix is required. For example: Carbon: 4 electrons, 4 basis functions normally (s, px, py, pz). For N atoms we have to know Nel/2 levels, that is 2N.In non-orthogonal TB re-orthogonalization also takes some time

In reality, we can relax a system composed from ~ 1000 atoms on a PC

The linear-scaling methods split into Variational methodsAnd moments methods: (based on the moment theorem, which relates the local electronic structure to the close environment. Works better for insulators

We can also use linear-scaling algorithms (additional approximation):

Basic idea: the force acting on atom depends mostly on positions/types of the close-by atoms

Review: S. Goedecker, Rev. Mod. Phys.,71 (1999) 1085

The power of simple tight-binding models

Assume that we know the atom positions (e.g., from experiment). Simple TB models make it possible:

Analytical calculations of the electronic structure (examples later on)

Numerical but highly efficient calculations of the Local Density of States (LDOS) without explicitly solving the Schroedinger equation. The recursion method is used instead

TB approximation

can be expressed in terms of the Green function

The main idea of the recursion method is to find a representation of the Hamiltonian where it has a three-diagonal term, than LDOS can be written as a chain fraction

For details, see Lecture 7 in http://www.acclab.helsinki.fi/~akrashen/esctmp.html


Predicting structural trends in solids

TB models can successfully be applied to predict (even analytically) structural trends in solids (i.e. which structure has lower energy)

Based on the famous structural energy difference theoremSee, e.g., D. Pettifor, Bonding and structure of molecules and solids

If the total energy can be represented as the sum of a bonding (electronic) and repulsive contributions , the structural energy difference U between two structures (molecule conformations, phases of bulk solids) is given to first order by

The proof of the theorem is trivial, if we suppose that the total energy is a function of some continuous parameter (nearest-neighbors bond length, volume of the unit cell, strain, etc) and expand the energy into Tailor series over this parameter. Details on the black board.

The theorem proved to work very well as shown by Pettifor within the context of TB model

0repUbondUU

repbond UUU


Sp2- hybridized carbonGraphene,the parentmaterial


Carbon nanotubes

S. Iijima, 1991

M. Endo, 1976


Understanding the electronic structure of carbon nanotubes

Experimental fact:Depending on how the nanotube is rolled up, the tube can be either metal or semiconductor

Let’s try to understand within the simple (only one parameter) TB model why some tubes are metals and other semiconductors!

Assume we know the atomic structure of the nanotube.


Electronic structure of grapheneAndre Geim and Kostja Novoselov (2004)

Let’s also understand why E ~ k using a simple TB (non-relativistic) model

Normally: *

2

2)(

mkkE

kvkE F)(Graphene:

Quoting from Physics Today: “The electrons in most conductors can be described by non-relativistic quantum mechanics, whereas the electrons in graphene need to be treated as relativistic particles called massless Dirac fermions”.


Solving the Schrödinger equation for an infinite chain

t is the same for all atom pairs!

..…1 2 3 4 N - 1 Na = 1The Bloch theorem:

1D2D

j > = ~ erj- r-R )j

jRj


One-band tight-binding approximation:

Electronic structure of a single graphene sheet

DO

S

Energy

-states -states

EF

b1

b2 b3

a1 a2

x

y +c2


b1

b2 b3

a1 a2

x

y

0

0t2

Recall: chain of atoms:

ikik

bkibki

eetk

eetk

cktckE

)(

)(

)()(21

11

After some elementary trigonometry



Semimetal (gapless insulator)!

(kx,k

y)

kK-point orDirac point


kvkkkkE Fyxyx ~~),( 22

If we expand near Dirac points

Electrons in graphene behave like photons and display properties quite different to those observed in other materials (anomalous quantum Hall effect, half integer quantization of Hall conductivity etc.)

Easy to show that

1. Density of states near Fermi energy EEg ~)(

2. There is a Van Hove singularity inthe density of states

(kx,k

y)

),( yx kkE

0,3

4K

Experiments on Angle-resolved photoemission confirm linear dispersion



From Graphene Sheets to Single WalledCarbon Nanotubes with Various Chiralities

the (10,10) armchair tube the (10,5) helical (chiral) tube

S. Maruyama, University of Tokyo


Electronic structure of carbon nanotubes

mkCh 2

C n a n ah 1 1 2 2{

Depending on the chiral indices n1, n2 a nanotube has either metallic or semiconducting properties!kx

ky

!

y

x

y

x

Bohr-Sommerfeld quantization

0,34

aK


Electronic structure of carbon nanotubes

The nanotube is metallic if:

or, equivalently:

Consider two non-equivalent K points

Why two conditions?

This result was obtained in 1992 by two different groups. No experimental information was available at that time!

What about experiments?

We must resolve atomic structure of an individual nanotube and its electronic properties, e.g., density of states.


Density of states and STM images of nanotubes

1998

Experimental STS characteristics

Theoretical DOS

TB approach catches the main physics of the electronic structure of nanotubes!

[C. Dekker, Delft Univ.]

L.C. Venema et al.,Phys. Rev. B 62 (2000) 5238.

For nanotubes with small diameters the picture is more complicated due to - orbital rehybridization

FYI


Graphene nanoribbonsGraphene-based nanoelectonicsProblem: graphene is a semi-metal: we cannot make transistors

Gap appears in graphene nanoribbons (GNR) of finite widths

However:

Gap depends on the width and edge type

FYI Simple TB (all hopping matrix elements are the same) give wrong values for the gap , but account for reconstruction of bonds at the edges gave correct results

Conclusions:Tight-binding approach is a good compromise between accuracy and computational efficiency

Density-functional-theory tight-binding method is particularly well developed; available as an independent program or integrated into many popular packages

TB (sometimes even analytical) calculations with very few adjustable parameters make frequently possible to get qualitative insight into the physics/chemistry of the system

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