Effective Electronic Structure Theory for Partially Non ... · Beyond clamped nucleus model: The Born-Oppenheimer (BO) and Born-Huang procedures Prepared by Shant Shahbazian for the

Effective Electronic Structure Theory for

Partially Non-Born-Oppenheimer Systems

Shant Shahbazian

Department of physics

Shahid Beheshti University

1Prepared by Shant Shahbazian for the Sixth Theoretical and Computational Chemistry Workshop at the Chemistry & Chemical Engineering Research Center of Iran (31-Jan-2017)

Main headlines and the organization of the lecture

•The electronic structure theory within the context of the clamped nucleus model

•The electronic structure theory beyond the context of the clamped nucleus model

•The “effective” electronic structure theory

•Prospects


The clamped nucleus model


ˆ ˆ ˆp p p

mol n nn

n n n n

H T V

1ˆ 0, 1 are electronsnT n

. . . .ˆ

elec elec elec elecH E

22

ˆ2

nNi

n n

in

Tm

.

1

q q

elec

Z ZU E

R R

. . . .ˆ

mol mol mol molH E

ˆn nN N

n nnn i j

i j n n

Q QV

r r

1 1 1 12

2

. 1

1 1 11

1ˆ2

N N N Nqi

elec i jii i i j i

ZH

m r rR r

1

1

0n

m

m

The consequences of the clamped nucleus model


1 21 1 1

. . 1 1 2 2

.

,..., , ,..., ,..., ,..., ,spin-variablesMN N N

mol mol M M

mol

r r r r r r

E numbers

11

. . 1 1 1

1

,..., ,electron spin-variables; ,...,

,...,

N

elec elec q

q

r r R R

U U R R

1,..., , the "potential energy surface" (PES),

is the cornerstone of chemistry

qU R R

What does a molecule look like beyond the clamped nucleus model?

5

• Since the PES disappears, concepts like molecular geometry, equilibrium structures, and reaction paths are generally vague

• Same nuclei are “indistinguishable” quantum particles like electrons and this is at variance with general intuition about nuclei

• Isomers share the same molecular Hamiltonian so where how one must deduce isomers in the molecular wavefunctions?

• How one may separate various types of motions from each other? How to disentangle translation, rotation, vibration, electronic from the molecular wavefunction?

• Where are chemical bonds?

• Molecular states are like “atomic” states…

• So, clamped nucleus model is a “paradigm”

Prepared by Shant Shahbazian for the Sixth Theoretical and Computational Chemistry Workshop at the Chemistry & Chemical Engineering Research Center of Iran (31-Jan-2017)

The electronic structure theory


. . . .ˆ How to solve?elec elec elec elecH E

ab initio methodolgies, semi-empirical methodologies, empirical methodologies, ...

Hartree-Fock(HF) Post-HF: MPn, CI, CC, MCSCF,explicitly correlated...ab initio methodolgies

Density Functional Theory (DFT) LDA, GGA, Hybrid, meta-GGA...

.Deriving approximate and PESelec

Beyond clamped nucleus model: The Born-Oppenheimer (BO) and Born-Huang procedures


. . . .ˆ

mol mol mol molH E

int . int . int . int . int . . .

int . .

2

translation continous spectrum

rotation

vibration

electronic

their couplingsThe BO procedure

ˆ ˆ ˆ ˆ,

ˆ ˆ ˆ

perturbation theory

mol CM

p

n elec

n

H E H H H

H T H

. . .ˆ , ... elec elec molH

. . . .

int . .,

ˆ

The Born-Huang expansion nuclear-variables

variational optimization

elec elec elec elec

i elec i

i

H E

What cannot be grasped within the clamped nucleus model?

8

• Non-adiabatic processes: Proton-coupled electron transfer (PCET)

• Bound states between usual molecules and exotic elementary particles like positrons and the positively charged muons


1, 0.005e e

positron muon

m m

m m

“Partially” non-BO (nBO) systems: Between the two extremes

9

• Certain nuclei are treated as quantum particles while others as clamped charges

• The PES loses its original “dimension”


ˆP nBO P nBO P nBO P nBOH E

1 21 1 1

1 1 2 2 1,..., , ,..., ,..., ,..., ,spin-variables; ,...,LN N N

P nBO P nBO L L Kr r r r r r R R

1

1

,...,K K

P nBO K P nBO

Z ZU R R E

R R

ˆ ˆ ˆp l p l p l

P nBO n nn

n n n

H T V

ˆ How to solve?P nBO P nBO P nBO P nBOH E

Nuclear-Electronic Orbital (NEO) theory

• “Enough” nuclei are clamped to ensure that no translational and rotational motions remains in the NEO Hamiltonian

• Thus, only the electronic and the nuclear vibrational motions and their couplings are described by the NEO Hamiltonian

• To each electron or nucleus (usually proton)/positron/muon a single-particle wavefunction (spin-orbital) is attributed

• The electronic spin-orbitals are similar to those used in the usual clamped nucleus based electronic structure theory while for the quantum nuclei they are used to describe mainly the vibrational motions


1 1 1 2 2 1 1, , , ,...., ,i j j i j j i j j

L Lr r r

“Hierarchical” structure of the NEO methodology


1 1 1 1 1

,

1

, ... ,1

... ... ...!

, ... ,

N

HF CI HF HF i SD i

i

N N N N N

r r

c cN

r r

HF Post-HF

mean field correlated (e-e correlation)

NEO-HF Post-NEO-HF

mean field correlated (e-e,e-n,n-n correlation)

1 2

1 1 1 1 1 2 2 2 2 2

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 2 2 2 2 2 2

1 2 1 1

1 1 1 1 1 1 2 2 2 2 2 2

, ... , , ... ,1

... ... ... ... ... ... ...! !...

, ... , , ... ,

N N

NEO HF

N N N N N N N N N N

r r r r

N Nr r r r

Basics of the NEO-HF theory


1

1

1 ,...,ˆ... , E=0LN

L NEO HF NEO NEO HF

spins

E dr dr H

1

1 1 1

1 1 1 1 1 1 1 1

1 1 1

2 2 2 2 2 2 2 2

/2 11 1 1 1 1

1 1 1 1 1 1 1 1 1

1 2

ˆ... , 1,..., 2

ˆ , 1,...,

....

ˆ ˆ ˆ ˆ ˆ 2

n

i i i

i i i

NN Lj j n

k

j n k

f r r r i N

f r r r i N

f r h r J r K r J r

1 2 2 1 2 2 2

1 1 1 1 1 1 1 1 1 1 11 211 1 11

1 1 2 2 2 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 11 2

1 1

1ˆ ˆ 2 , ,

1 1ˆ ˆ ,

Kj j j

j i j i j n k

k n n n

Zh r m J r dr r r

r rr R

K r r dr r r r J r dr rr r

1

1 1

1

similarly for other quantum particles

k

n n

n

rr r

Electronic and nuclear basis sets


1

1 1, , and are gaussian functionsnMM

i i i i i i

m m n m m

m m

c f d g f g

• Electronic basis sets are those used in the usual clamped nucleus based electronic structure theory describing delocalized electronic orbitals

• Nuclear basis sets are used to describe vibrational motion of the quantum nuclei and are usually composed of the s-, p-, and d-type basis functions

• Using electronic and nuclear basis sets the NEO-HF equations are transformed to their algebraic version and then can be solved straightforwardly by the SCF procedure

• The resulting wavefunction is the starting point for many post-NEO-HF methods that deal mainly with e-e and e-n correlations (n-n correlation is usually negligible)

Simplifying the NEO-HF theory: The Hartree product wavefunction

• In 2010 it was proposed that the nuclear wavefunction may be simplified considerably using the Hartree product wavefunction

• The justification behind using this wavefunction is the “localized” nature of nuclear orbitals that prevents “effective overlap” between nuclear orbitals making them practically “distinguishable” particles

• It was shown that the computational cost of the ab initio calculations significantly is reduced using this new form of the equations


1 11 1

1 1 1 1 1

2

( , ,..., , ) ( )p

N N

SD HP n n n

n

r r r

1

1

/21 1 1 1 1

1 1 1 1 1 1 1 1 1

1 2

2

1

2, 1

ˆ ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ ˆ2

N pj j

n

j n

Npi

n n n n l n n

l l n i

f r h r J r K r J r

f r h r J r J r

“Pseudo-adiabatic” reinterpretation of the Hartree product wavefunction

• Each nucleus may be conceived as a “quantum oscillator” in a hypothetical external field while with such interpretation the nuclear orbital is eigenfunction of the oscillator

• The simplest model for this oscillator is the isotropic harmonic oscillator with two unknown parameters

• Interestingly the ground state eigenfunction of this model is a s-type gaussian function, which has been used as a nuclear basis set in ab initio calculations as well


2

2 .ˆ ˆ ˆ, 2

ocs ocs ext

n n n n n n n n n n

n

H r e r H V rm

2 22

.

,

2ˆ ext nn n n c

n

V r Rm

3

4 2

,

2( ) expn

n n n n n cr r R

Deriving the effective electronic Hamiltonian


1 11 * 1

1 1 1 1 1 1 1

2 2

ˆ

ˆ( ... ) ( ... )

eleceff

p pN N

n n n NEO n n

n n

H

d r r d r H r r r

1 1 1 12

1

22.

1

1 1 1 1 21 1 1 1

1 1

2 21 , ,

2 1 2 11 , ,

31ˆ2 2

2 2

N N N N q pelec i neff i j i

i i j i i n n

Np p qin n

n n c n n cin i nn c n c

n l

ZH

m mr r R r

Q Q Zerf r R erf R R

r R R R

Q Q

R

1

2

, ,

2 , ,

2p p

n ln c l c

n l n n ln c l c

erf R RR

More on the effective electronic Hamiltonian

Prepared by Shant Shahbazian for the Sixth Theoretical and Computational Chemistry Workshop at the Chemistry & Chemical Engineering Research Center of Iran (31-Jan-2017) 17

1 1 1 12

2

. 1

1 1 1 2 11 1 1 11

1ˆ2

N N N Nq pBO i nelec i i ji

i i n i j in

Z QH

m r r r rR r

2 2 1

p p p q

n l nclassic

n l n nn l n

Q Q Q ZV

r r R r

.

.ˆ ˆlim

n

elec

m eff elec classicH H V

1 ,, ,0

n n cR

The “effective” HF equations and their implications


1 1 1

/2

1 1 1 , 1 1

2 1 ,

22

1 1

1 1

ˆ

1ˆ ˆ ˆ ˆ 2 2

ˆ 2

e

eff

i i i

Npeff

n n c j j

n jn c

q

f r r r

f r h r erf r R J r K rr R

Zh r

m r R

1

12 2

2, , ,

2 2 1 2, , ,

23 2

2

p p q p p

n n l n ln n n c n c l c

n n n l nn n ln c n c l c

Q Z Q QU erf R R erf R R

m R R R R

1 2 2, ,,..., , , ,..., , , Adding to the dimension of the PES!K c n n cU R R R R

Beyond the isotropic harmonic model of nuclei

• It is possible to assume an anisotropic anharmonic oscillator model for nuclei and in that case the effective electronic Hamiltonian will be more complicated mathematically (arXive:1611.07960)

• A new set of the effective HF equations are derived for this effective electronic Hamiltonian

• The computational implementation of the effective HF equations is quite straightforward and the existing HF computer codes may easily be re-programmed to solve the effective HF equations

• The computer code may be used to study molecular systems containing quantum protons or other isotopes of hydrogen as well as the positively charged muons

• The same procedure maybe used to deduce the “effective” Kohn-Sham equations within context of the “effective” DFT


Numerical examples: HCN and MuCN


Prospects

• A DFT is derivable using the effective electronic Hamiltonian by restating the two basic Hohenberg-Kohn theorems

• This opens the door also for introducing a novel “conceptual” DFT, which goes beyond the clamped nucleus model enable to capture subtle isotope effects

• The effective electronic Hamiltonian may be used as a starting point to deduce novel wavefunction based ab initio procedure to recover e-e correlation using more complicated electronic wavefunctions than the single determinant

• However, the effective electronic Hamiltonian totally lacks any e-n correlation because of its product nature thus unable “intrinsically” to recover this type of correlation which is by no means negligible

• The most interesting possible extension of the effective theory is proposing a new type of “minimal” wavefunction that contains the e-n correlation and to be used as a new starting point to derive a novel effective electronic Hamiltonian containing an effective potential capable of reproducing the e-n correlation effects on electrons properly


References for further reading

[1] S. P. Webb, T. Iordanov, S. Hammes-Schiffer, J. Chem. Phys. 117, 4106 (2002).

[2] T. Iordanov, S. Hammes-Schiffer, J. Chem. Phys. 118, 9489 (2003).

[3] B. Auer, S. Hammes-Schiffer, J. Chem. Phys. 132, 084110 (2010).

[4] M. Gharabaghi, Sh. Shahbazian, Phys. Lett. A 380, 3983 (2016).

[5] M. Ryaka, M. Goli, Sh. Shahbazian, arXive:1611.07960

[6] M. Ryaka, M. Goli, Sh. Shahbazian, under preparation.


Thanks for your attention


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Effective Electronic Structure Theory for Partially Non ... · Beyond clamped nucleus model: The Born-Oppenheimer (BO) and Born-Huang procedures Prepared by Shant Shahbazian for the