phd defence part 2

Second Order Generalized Van Vleck Perturbation Theory Molecular Gradients

and Nonadiabatic CouplingTerms

Daniel P. TheisUniversity of North Dakota

Chemistry DepartmentGrand Forks, ND

2

• Introduction

Outline

Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133.

• GVVPT2 Gradiengts

• GVVPT2 Nonadiabatic Coupling Terms

3

1. GVVPT2 is a post-MCSCF, perturbation based electronic structure method.

Benefits of the GVVPT2 Method


2. GVVPT2 can determine accurate electronic energies for systems with several low lying, nearly degenerate electronic states.

3. GVVPT2 potential energy surfaces are continuous, differentiable functions of the geometry, that ensure the evaluation of molecular gradients.

4. A highly efficient algorithm has been constructed for the GVVPT2 method, which combines the advantages of the macroconfiguration and GUGA techniques.

4

t

Ltt

MRCI AFT

HMM

HQM

HMQ

HQQ


MCSCF

The GVVPT Methodology

◦ Relatively Small CSF space, LM

◦ Neglects Dynamic Correlation (LQ)

mILm

mMCI CF

M

◦ Large CSF Space, LT = LM LQ

◦ Includes Dynamic Correlation.

MRCISD

0

MLm

mImmMCImm CEH

0

TLtttt

MRCItt AEH

5

MLmmm

GV AF

HQM

HMQ


effMMH

GVVPT Methods


HQQ

,ˆ GVVGVV

,0

MLmmmm

GVVeffmm AEH

mmeffmm FHFH

ˆˆˆ

◦ Heff is the same dimension as HMM

◦ Perturbative corrections from HQM, HMQ, and the diagonal elements of HQQ add dynamic correlation.

6

effQPH

effPQH

effPSH

effSPH

effPPH

effSSH

effSQH

effSQH

effQQH


Shavitt, I.; Redmon, L. T. J. Chem. Phys. 1980, 73, 5711.Kuhler, K.; Hoffmann, M. R. J. Math. Chem. 1996, 20, 351.Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133.

The effective Hamiltonian is formed by:

• LM = LP ⊕ LS.

7

effQPH

effPQH

effPSH

effSPH

effPPH

effSSH

effSQH

effSQH

effQQH

0HH XX QP

effQP ee



Xeˆˆ

QPQPPQPQX FXFFXF ˆ


• LM = LP ⊕ LS.

• Define Ω, X, and XQP such that:ˆ ˆ

◦

◦

◦

8

effQPH

effPQH

effPSH

effSPH

effPPH

effSSH

effSQH

effSQH

effQQH

• Perturbatively expand Heff and X.

0HH XX QP

effQP ee



Xeˆˆ

QPQPPQPQX FXFFXF ˆ


• LM = LP ⊕ LS.

• Define Ω, X, and XQP such that:ˆ ˆ

◦

◦

◦

)2()2(21)2(

MMMMMMeffMM ZZHH

PMQPMQPMMPMMM CXHCCIZ )1()2( 2

GVVPT2 Effective Hamiltonian

9Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133.

The Origen of GVVPT2’s Stable Potential Energy Curves

)1(qIX

I

em

a

2a

3a

4a

-4a

-3a

-2a

-a

a = HqI

I

Iq

II

qIqI

ee

HHX

mm

0

)1(

QIQMCIQQQI E HIHX

1)1(

Kuhler, K.; Hoffmann, M. R. J. Math. Chem. 1996, 20, 351.

10Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133.

)1(qIX

I

em

a

2a

3a

4a

-4a

-3a

-2a

-a

a = HqI

I

IqqI

eE

HX

m)1(

e

eeeLq

qIIII HE

m

xxxx mmm22

41

21 )()()()( ◦

QIQMCIQQQI E HIHX

1)1(



11

qII

qI HDXem)1(


I

II

e

e

e E

ED

m

mm

tanh

)1(qIX

I

em

a

2a

3a

4a

-4a

-3a

-2a

-a

a = HqI

◦

QIQMCIQQQI E HIHX

1)1(



12

where each j is determine from

…

e1(x;(x)) = 0e2(x;(x)) = 0E = E(x;(x))

Determining Energy Gradients

Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.

13

j a

j

jaa x

E

x

E

dx

dE

)(; xλx

where each j is determine from

…

e1(x;(x)) = 0e2(x;(x)) = 0E = E(x;(x))

0

)(;

j a

j

j

i

a

i

a

i

x

e

x

e

dx

de

xλx

a

j

x

where each is determined from

Determining Energy Gradients


14

i

ii eEL )(;)()(;)(),(; xλxxxλxxξxλx

0

j

L

0)(;

xλxi

i

eL

where the values for i and j are determined by requiring

and

The Lagrangian Approach of Determining Energy Gradients


15


i


0

j

L

0)(;

xλxi

i

eL


and

Under these conditions

aaa x

L

dx

dL

dx

dE

)(; xλx


16


i


0

j

L

0)(;

xλxi

i

eL


and

Under these conditions

aaa x

L

dx

dL

dx

dE

)(; xλx


n

n

n

n

nn

n

n

E

E

E

eee

eee

eee

2

1

0

01

01

21

22

2

2

1

11

2

1

1

)(

)(

)(

x

x

x

17

MLmMP

rveffmmmm

GVV HAAE )(,)(,)(;)()()( xCxΔxΔxxxx•

Electron Structure Parameters that are used in the GVVPT2 Calculations

18

MLmMP

rveffmmmm



MLm

mIrv

mMPrvMC

I CF )()(,)(;)(,)(,)(; xxΔxΔxxCxΔxΔxCmI(x) ~•

19

MLmMP

rveffmmmm



MLm

mIrv

mMPrvMC


~)(and )( xx abr

ijv

iGk

kiomok

rvmoi )(exp)()(,)(; xΔxxΔxΔx

v

r

ik kGG Gk

kiomok )(exp)( xΔx

•

•iGiNote:

20

MLmMP

rveffmmmm



MLm

mIrv

mMPrvMC


~)(and )( xx abr

ijv

iGk

kiomok

rvmoi )(exp)()(,)(; xΔxxΔxΔx

v

r

ik kGG Gk

kiomok )(exp)( xΔx

•

•iGiNote:

OMO

ijklb

omoijklA

omoijkl

OMO

ijB

omoijA

omoijAB FeFgFEFhH

~ˆ

~)(

~ˆ~)()( 2

1 xxx

21

~ 1)( 2 MLm

mIC x 0)(

MLm

ImMCImmmm CEH xand

The Constraint Equations that are used by the GVVPT2 and MRCI Calculations

CmI(x)•

22

0)()(1)( 2

M M

LmIm

LmIm

MCImmmmmI CCEHe xxx CmI(x)• ~


23

0)()(1)( 2

M M

LmIm

LmIm

MCImmmmmI CCEHe xxx CmI(x)•

~)(xijv••

~

0)(ˆˆ),()()( xxxx MCIjiij

MCI

N

IIij EEHwe

P


24

0)()(1)( 2

M M

LmIm

LmIm

MCImmmmmI CCEHe xxx CmI(x)•

~)(xijv••

~


)(xabr ~ 0)()()()()( 2

1 OCC

klakblabklklababab gghfe xxxxx •

0)(ˆˆ),()()( xxxx MCIjiij

MCI

N

IIij EEHwe

P

25

PN

I

MO

i

aIii

GVViI

MCI

GVVMO

ij

aijkl

GVVijkl

MO

ij

aij

GVVij

a

GVV

fPghx

E21•

PN

I

MO

k ijv

IkkGVV

kIMCI

GVV

MO

xyzjxyz

GVVixyz

MO

xjx

GVVix

MO

xyzixyz

GVVjxyz

MO

xix

GVVjx

ijv

GVV

fP

ghghE

21

2222

Efficiently Evaluating the Partial Derivatives of EGVV

•

26

PN

I

MO

i

aIii

GVViI

MCI

GVVMO

ij

aijkl

GVVijkl

MO

ij

aij

GVVij

a

GVV

fPghx

E21•

PN

I

MO

k ijv

IkkGVV

kIMCI

GVV

MO

xyzjxyz

GVVixyz

MO

xjx

GVVix

MO

xyzixyz

GVVjxyz

MO

xix

GVVjx

ijv

GVV

fP

ghghE

21

2222

MIPMMIPMMCP

GVVMIMm

PmMCP

GVVQ

mI

IQMCI

GVV

mI

GVV

A

CC

E

CHXHXCΦAHX

HXΦHβX

212


•

•

27

Qe e M

P

e

L q Lm

N

IqjiijmMIQMqmI

GVVij FEEFW

m m

m CH ˆ21

•


Qe e M

P

e

L q Lm

N

IqjilkjiklijlkijklmMIQMqmI

GVVijkl FeeeeFW

m m

m CHˆˆ

81

Qe e

e

L qmqMIQMqImI HX

m m

m CHHX

Qe e

e

L qmqMIQMqIQ

mI

IQ HVC m m

m CHHβX

GVVPT2 Energies – Slowest Step

GVVPT2 vij, rab, and CmI Derivatives – Slowest Steps

•

•

•

~ 2 × time of (HX)mI

~ 1 × time of (HX)mI

28

Molecule Analytical Values Deviation from Numerical Values

Geometry Description X Y Z X Y Z

H2CO C -0.171726 0.205990 0.000000 -6.90×10-7 -8.00×10-8 0.00

CH1 Str. (+0.5 Å) O 0.081525 -0.025708 0.000000 3.70×10-7 4.20×10-7 0.00

H1 0.240019 -0.424267 0.000000 -6.00×10-8 7.00×10-8 0.00

H2 -0.149818 0.243985 0.000000 9.00×10-8 -1.20×10-7 0.00

LiH (X 1+) H 0.000000 0.000000 -0.014113 0.00 0.00 0.00

Avoided Crossing Li 0.000000 0.000000 0.014113 0.00 0.00 0.00

LiH (A 1+) H 0.000000 0.000000 -0.004441 0.00 0.00 0.00

Avoided Crossing Li 0.000000 0.000000 0.004441 0.00 0.00 0.00

• cc-pVTZ Basis Set

Technical Details: H2CO (Cs – Broken Sym.)

• RCO = 1.205 Å, RCH = 1.611 Å, and RCH = 1.111 Å.

• 2 Val. Groups {612 40; 611 41; 610 42}

HCH = 116.1o and OCH = 121.9o1 2

• Roos Aug. TZ Basis Set

Technical Details: LiH (C∞v)

• 9:1 SA-MCSCF MOs

• (2:10)-CAS + 1 Core Orb.

• RLiH = 3.400 Å

Analytical GVVPT2 Gradients for H2CO and LiH

29

The Lagrangian Approach of Determining Nonadiabatic Coupling Terms

i

ii eg )(;)()(;)(),(; xλxxxλxxξxλx L

where g'(x) is a function that makes

0

jL

andSetting 0

iL

, makes:

aaaaa xdx

d

dx

dg

dx

d

dx

d

)()()()()(

21 xxxxx LL

aaa dx

d

dx

d

dx

dg )()()(21 xxx

30


i



MeffMMGVGVMMPMMPPMMPM

GV

EECCg AxHxFFxxAx )(

1)()()()( 2

1

aaa dx

d

dx

d

dx

dg )()()(21 xxx

For GVVPT2

31


i



MeffMMGVGVMMPMMPPMMPM

GV

EECCg AxHxFFxxAx )(

1)()()()( 2

1

aaa dx

d

dx

d

dx

dg )()()(21 xxx

For GVVPT2

)()()(ˆ)( )2()2( xAxFxx MMGV

• Geometry optimizations, gradients calculations, and frequency calculations verify that the GVVPT2 method accurately describes the chemically important regions of most potential energy surfaces.

Conclusions

32

• The GVVPT2 gradients are continuous across potential energy surfaces, including regions of avoided crossings.

• Analytic formulas for GVVPT2 molecular gradients and nonadiabatic coupling terms have been developed which scale at approximately 2-3 times the speed of the GVVPT2 energy.

• Computational implementation of GVVPT2 analytic gradients show excellent agreement with finite difference calculations.

Dr. Mark R. HoffmannDr. Yuriy G. Khait

Patrick TamukangRashel MokambeJason HicksErik Timmian

Dr. Theresa WindusDr. Klaus Ruedenberg

Acknowledgements

33

phd defence part 2

Documents

gvvpt2 method hoffmann

hqm hmq hqq hoffmann

gvvpt2 effective hamiltonian

introduction outline

gvvpt methodology hqq

gvvpt methodology shavitt

gv af hqm hmq hoffmann

qi qi ee hh x