A Gentle Intro to Tight-Binding Modeling

Dr. Walczak’s Group Meeting 2017

“A Gentle Intro to Tight-Binding Modeling” OBJECTIVES

 To justify tight-binding modeling by building it on variational method in quantum mechanics.  To show how superposition of states and quantum interference effects create bonding, non-bonding, and anti-bonding orbitals.  To explain the concepts of singlet and triplet states, orbital hybridization, and Hückel theory.

“Organometallic Chemistry”

Dr. Kamil Walczak Department of Chemistry and Physical Sciences, Pace University 1 Pace Plaza, New York City, NY 10038

Dr. Walczak’s Research Group Principal Investigator: Dr. Kamil Walczak Student:

Topics:

Rita Aghjayan

“Thermal Rectifiers”, “Relaxation Processes”

Arthur Luniewski

“Molecular Noise”, “High-Intensity Heat Fluxes”

David Saroka

“Tunneling of Heat”, “Wave-Packet Approach”

Joanna Dyrkacz

“Inelastic Heat Flow”, “Electron-Phonon Effects”

Luke Shapiro

“Thermal Memristors”, “Hysteretic Loops”

Hunter Tonn

“Quantum Interference”, “Tunneling of Heat”

Erica Butts

“Neuronal Networks”, “Memristive Ion Channels”

Omar Tsoutiev

“Thermal Noise”, “Quantum-Interference Devices”

Kamil Walczak

10 / 31 / 2017

Approximate Methods in Quantum Mechanics Approximate Procedures Variational Method

Perturbation Theory

Requires a reasonable trial wavefunction (experience in making good guesses)

Requires a solution for simpler problem (simplified form of Hamiltonian)

Useful in evaluations of ground states (limited applicability to excited states)

Applicable to small perturbations (perturbation brings small correction)

Useful approach to many-particle systems (e.g. quantum chemistry)

Useful approach to interactions with external fields (e.g. quantum optics)

Kamil Walczak

10 / 31 / 2017

Variational Method For any trial wavefunction, the expectation value of energy is always larger or equal to the true ground-state energy of the system under consideration:

(Rayleigh Ratio) =

 ˆ   t Ht dV



 t

t dV

(integral representation)



ˆ  t H t t t

 E gs

(Dirac notation)

The variational method is an effective tool for ground-state energy evaluation via minimization of energy condition by using reasonable trial wavefunctions! In further analysis, we assume that trial wavefunctions are normalized:    t t dV  t t  1

Kamil Walczak

10 / 31 / 2017

Variational Principle: Proof According to Born theorem, the trial wavefunctions can be expressed as a linear combination of orthonormal eigenfunctions of analyzed Hamiltonian: 

t   a m m

ˆ E  H m m m

where

m 1

Consider the following expression:





ˆ  E  dV   H gs t   t









  ˆ c c   m n  m H  Egs n dV

m , n 1



2   c c E  E   dV  | c |   n E n  Egs   0 n gs  n

m , n 1







 m n

 m

n 1

 mn

0

0

ˆ  E  dV  0    H ˆ  dV  E   t H gs t t t gs  Kamil Walczak

10 / 31 / 2017

Ritz Method (1909) Variational principle is defined via the so-called Rayleigh ratio:



ˆ  t H t t t

 E gs

Ritz trick: trial wavefunction is a linear combination of N known basis functions φ (like Gaussian, Slater, Lorentzian, etc.): N

t   ci i i 1

Collecting all terms on one side, we obtain N secular equations:

 c H N

j1

j

kj

Sij  i  j Kamil Walczak

  Skj   0 Overlap integrals

k  1,2,3,..., N with , {c j} unknown for

ˆ  Hij  i H j

Walther Ritz Swiss Physicist (1878-1909)

Hamiltonian elements 10 / 31 / 2017

Secular Equations Let us differentiate both sides of previous equation with respect to conjugated expansion coefficients: N

N

 i 1 j1

N N ci ci c jSij   c jH ij   c k i 1 j1 c k

ik

ik

Collecting all terms on one side, we obtain N secular equations:

 c H N

j1

j

kj

k  1,2,3,..., N with , {c j} unknown

  Skj   0

for

This is a homogeneous set of linear equations with respect to column vector c. Non-trivial solution is expected when the appropriate determinant vanishes:





det H   S  0 Kamil Walczak

10 / 31 / 2017

Hydrogen Molecule: Tight-Binding Secular equations for hydrogen molecule:

 E H    U   S c A      0  U   S E H    c B  Trial wavefunction (LCAO method):

S (r1 , r2 )  cA A (r1 )  cBB (r2 )

A (r )

V( r )

B (r )

V( r )

V( r ) p  ( A)

p  (B)

Nomenclature (parameterization):

ˆ    H ˆ  E H  A H A B B

A A  B B  1 ˆ    H ˆ  0 U   A H B B A

S  A B  B A  0,1 Kamil Walczak

(on-site electronic energy)

(normalization condition) (intersite el-el interaction) (overlap integral) 10 / 31 / 2017

Hydrogen Molecule: Energy Levels Solving quadratic equation by making secular determinant equal to zero:

 E H    U   S det  0  U   S E H   

EH

2U

EH  U  1 S

(bonding state energy)

(anti-bonding state energy)

1 A (r)  B (r) A  2

  EH  U

EH

EH  U  1 S

Molecular Orbitals

  EH  U 1 A (r)  B (r) S  2 Kamil Walczak

10 / 31 / 2017

Molecular Orbitals in-phase

bonding molecular orbital out-of-phase

anti-bonding molecular orbital Kamil Walczak

10 / 31 / 2017

Crucial Parameters

E

Dissociation energy:

E A (R ) R 2E H

E S (R )

D  4.52 [eV]

TB parameters for hydrogen molecule:

Interatomic separation:

E H  13.6 [eV]

R eq  0.074 [nm]

U  18.12 [eV]

S0

Ground-state energy for hydrogen molecule:

E gs  31.72 [eV] Kamil Walczak

10 / 31 / 2017

Addition of Angular Momenta Two spin-1/2 particles – four basis states:









1 1  

1 0     / 2

s 1

(triplet combination)

s0

(singlet combination)

1 1  

0 0     / 2

Almost all molecules encountered in daily life exist in a singlet state! (oxygen diatomic molecule is one exception) Kamil Walczak

10 / 31 / 2017

Spins and Statistics Spatial wavefunctions for indistinguishable quantum particles:

S r1 , r2   NA (r1 )B (r2 )  B (r1 )A (r2 )  S r2 , r1  (symmetric wavefunction for bosons = integer spin)

A r1 , r2   NA (r1 )B (r2 )  B (r1 )A (r2 )  A r2 , r1  (anti-symmetric wavefunction for fermions = half-integer spin)

To obtain the overall anti-symmetric wavefunction for two electrons:

Kamil Walczak

S r1 , r2  

(singlet)

A r1 , r2  

(triplet) 10 / 31 / 2017

Lithium Hydride Solving quadratic equation for determinant being equal to zero:

U  E H   det  0    U E Li  

Li

H

E H  E Li (E H  E Li ) 2    U2 2 4 (bonding state energy)

E H  E Li (E H  E Li ) 2    U2 2 4 (anti-bonding state energy) Kamil Walczak

10 / 31 / 2017

Covalency and Polarity HOMO-LUMO gap:

E g  (E H  E Li ) 2  4U 2 Ionic contribution (electronegativity)

Covalent bonding contribution (overlapping)

HOMO – Highest Occupied Molecular Orbital LUMO – Lowest Unoccupied Molecular Orbital

Electronegativity is a general tendency of an atom or a functional group to attract electrons (shift electron density) towards themselves! (covalency)

C  Kamil Walczak

2U (E H  E Li ) 2  4U 2

(polarity)

P 

| E H  E Li | (E H  E Li ) 2  4U 2 10 / 31 / 2017

Li-H: Molecular Orbitals Molecular orbitals related to bond polarity parameter:

S 

1  P 1  P H (r )  Li (r ) 2 2

1 P 1  P A  H (r )  Li (r ) 2 2 For all bonds:

 C2   2P  1 Sigma bond (“σ”) – the strongest type of covalent bonds, formed by head-on overlapping between atomic orbitals, electrons strongly localized within bond! Pi bond (“π”) – weak covalent bonds, formed by overlapping between two lobes of two atomic orbitals involved, electrons are highly delocalized (spread all over)! Kamil Walczak

10 / 31 / 2017

Dirac Statement (1929) “The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved!”

Molecular Hamiltonian (+ Born-Oppenheimer Approximation):

Kamil Walczak

Paul Dirac (1902-1984)

10 / 31 / 2017

Orbital Hybridization (1931) Orbital hybridization – the concept of mixing atomic orbitals to form new hybrid orbitals suitable for qualitative description of atomic bonding and explanation of the shape of molecular orbitals for molecules!

Linus Pauling American Chemist (1901-1994)

Kamil Walczak

10 / 31 / 2017

Hückel Method (1931) Hückel Molecular Orbital (HMO) theory – semi-empirical method used to determine stability, energy levels, and approximated molecular orbitals of delocalized “π” electrons in aromatic and conjugated organic molecules! Sigma-pi separability – since “sigma” and “pi” orbitals in planar molecules are orthogonal, both subsystems are separable! Negligence of sigma electrons – since only “pi”-type electrons determine most properties of organic molecules, skeleton electrons of “sigma”-bonds may be neglected!

Linear Combination of Atomic Orbitals (LCAO) – where one electron described by “2pz” per carbon atom is included: N

t   ci i

where

i  2p z ri 

i 1

Erich Hückel German Physicist (1896-1980)

E. Hückel, Z. Phys. 70, 204 (1931); 72, 310 (1931) Kamil Walczak

10 / 31 / 2017

Hückel Method: Parametrization Secular equations in the Rayleigh-Ritz variational method:

 H N

k 1

jk

  S jk  c k  0

Hückel parameters:

ˆ  H jk   j H k

S jk   j k

   6.6 [eV ] for j = k (Coulomb integral)      2.7 [eV ] for bonded j and k (resonance integral)  0 otherwise 

1 for j = k (overlap integral)  0 otherwise

Procedure: first we need to evaluate the values of the parameter ε by making secular determinant equal to zero to determine the eigenvalues of energies. Then we need to solve a homogeneous set of linear equations for coefficients “c” with given energies to determine the corresponding molecular orbitals! Kamil Walczak

10 / 31 / 2017

Ethylene (Ethene)

A

B

H 2C  CH 2

Trial wavefunction (LCAO method):

t  cA A  cBB Secular equations for ethylene molecule:

      c A        c   0   B  Solving quadratic equation by making secular determinant equal to zero:

      det  0        Kamil Walczak

    ,

    10 / 31 / 2017

Ethylene: HMO Solutions

π*-orbital (antibonding MO)

1 A (r)  B (r) A  2

   

2p z

2p z

2    

π-orbital (bonding MO)

Kamil Walczak

Molecular Orbitals

1 A (r)  B (r) S  2 10 / 31 / 2017

Ethylene: Response to Twisting “Loss of antibonding overlap” (stabilization of π* orbital)

π*-orbital (antibonding MO)

π-orbital (bonding MO)

“Loss of bonding overlap” (destabilization of π orbital) Kamil Walczak

10 / 31 / 2017

Allyl Anion

Resonance structures (canonical forms; mesomerism) Trial wavefunction (LCAO method):

t  cA A  cBB  cCC Secular equations for allyl anion:

0  c A                c   0   B   0      c C  Kamil Walczak

10 / 31 / 2017

Allyl Anion: Solutions 3  0.5 A  0.7 B  0.5 C

  2 

π*-orbital (antibonding MO)

 2.78 [eV] nodes SOMO

  6.6 [eV]

  2 

 10.42 [eV]

2  0.7 A  0.7 C n-orbital (non-bonding MO)

3  0.5 A  0.7 B  0.5 C π-orbital (bonding MO)

SOMO – Singly Occupied Molecular Orbital Kamil Walczak

10 / 31 / 2017

1,3-Butadiene

C

A B

D

H 2C  CH  CH  CH 2

Trial wavefunction (LCAO method):

t  cA A  cBB  cCC  cDD Secular equations for butadiene molecule:

0 0  c A               0  c B   0    0         cC    0       c D   0 Kamil Walczak

10 / 31 / 2017

Butadiene: Electronic Structure 5 1     2.23 [eV ] 2

LUMO

5 1     4.93 [eV ] 2 5  1  1.24 [eV]

HOMO

 

(HOMO-LUMO gap)

5 1   8.27 [eV] 2

5 1     10.97 [eV ] 2 Kamil Walczak

10 / 31 / 2017

Butadiene: Molecular Orbitals Antibonding molecular orbitals:

4  0.37 A  0.60 B  0.60 C  0.37 D 3  0.60 A  0.37 B  0.37 C  0.60 D

x  2p z rx 

where

x  A, B, C, D.

Bonding molecular orbitals:

2  0.60 A  0.37 B  0.37 C  0.60 D 1  0.37 A  0.60 B  0.60 C  0.37 D Kamil Walczak

10 / 31 / 2017

Cyclobutadiene (Lifetime ≈ 5s) A

D

B

C

CH  CH

CH  CH

CH  CH

CH  CH

Resonance structures (canonical forms; mesomerism)

Trial wavefunction (LCAO method):

t  cA A  cBB  cCC  cDD Secular equations for cyclobutadiene molecule:

0    c A               0  c B   0    0         cC    0       c D    Kamil Walczak

10 / 31 / 2017

Cyclobutadiene: Electronic Structure

    2  1.2 [eV]

HOMO = LUMO

    6.6 [eV]

    2  12.0 [eV]

Kamil Walczak

10 / 31 / 2017

Cyclobutadiene: Molecular Orbitals Antibonding molecular orbitals:

4  0.5 A  0.5 B  0.5 C  0.5 D Non-bonding molecular orbitals:

3  0.5 A  0.5 B  0.5 C  0.5 D

x  2p z rx 

where

x  A, B, C, D.

Non-bonding molecular orbitals:

2  0.5 A  0.5 B  0.5 C  0.5 D Bonding molecular orbitals:

1  0.5 A  0.5 B  0.5 C  0.5 D Kamil Walczak

10 / 31 / 2017

Delocalization Energy: Molecular Stability



Total energy of “π” electrons in butadiene:







E   2  5  1   2  5  1   4  2 5   38.5 [eV]







Edel  22  2  4  2 5   2 5  4   1.27 [eV] Total energy of “π” electrons in cyclobutadiene:

E   2  4  2  4  4  37.2 [eV]

Edel  22  2  4  4  0 Kamil Walczak

10 / 31 / 2017

Benzene A

C  C : 154 [pm]

B

F

C  C : 133 [pm]

C E

D

Resonance structures (canonical forms; mesomerism) Secular equations for benzene molecule:

0 0 0    c A               0 0 0  c B    0     0 0  c C      0 0     0  c D   0    0 0 0         c E     0 0 0      c F     Kamil Walczak

10 / 31 / 2017

Benzene: Electronic Structure     2  1.2 [eV]

      3.9 [eV] 2  5.4 [eV]

(HOMO-LUMO gap)

      9.3 [eV]

    2  12.0 [eV] Kamil Walczak

10 / 31 / 2017

Benzene: Molecular Orbitals 6 

1  A  B  C  D  E  F  6 1 5   A  C  D  F  2 1 4   A  B  D  E  2

1 A  C  D  F  2 1 2   A  B  D  E  2 1 A  B  C  D  E  F  1  6 3 

Kamil Walczak

10 / 31 / 2017

Homework Assignment Use Hückel method to calculate and analyze: (a) the accessible energy levels, (b) molecular orbitals, and (c) delocalization energy for pentadiene and cyclopentadiene!

Kamil Walczak

PENTADIENE

CYCLOPENTADIENE (Stable)

H 2C  CH  CH  CH  CH3

C5 H 6 10 / 31 / 2017

Organometallic Transport Junction

S

Cyclopentadiene Kamil Walczak

S

Au Contact

Au Contact

Since cyclopentadiene belongs to stable organic compounds, it may be used to build organometallic compounds which contain chemical bonds between carbon atoms and metal atoms (alkaline, alkaline earth, transition metal, etc.)! Those organometallic compounds may be connected to gold electrodes via sulfur atoms (thiol groups) through which transport may occur!

Fe, Co, Mg Ti, Mo, Cr 10 / 31 / 2017