Comparison of Born-Oppenheimer and Atom-Centered Density Matrix

Propagation Methods for Ab Initio Molecular Dynamics

H. Bernhard SchlegelDept. of Chemistry, Wayne State U.

Detroit, Michigan 48202, USA

• Born-Oppenheimer Dynamics– to move on an accurate PES, converge the

electronic structure calculation at each point

• Extended Lagrangian Dynamics– propagate the electronic structure as well as

nuclei using an extended Lagrangian (e.g. Car-Parrinello, ADMP, etc.)

Approaches for

Ab Initio Molecular Dynamics

• Born-Oppenheimer approach– to get an accurate PES at each step, converge

wavefunction rather than propagate– gradient-based integrators

• standard numerical methods• small step sizes

– Hessian-based integrators• local quadratic surface - larger steps

– Helgaker et al. CPL 173, 145 (1990)

• predictor – corrector based on a local 5th order surface– CPL 228, 436 (1994), JCP 111, 3800 and 8773 (1999)

Ab Initio Classical Trajectory on theBorn-Oppenheimer Surface Using a

Hessian-based Predictor-Corrector Method

Calculate the energy,gradient and Hessian

Solve the classicalequations of motion on a

local 5th order polynomial surface

Energy Conservation as a Function of Step Size for Hessian Based Integrators

-12

-10

-8

-6

-4

-2

0

-2.5 -2 -1.5 -1 -0.5 0Logarithm of mass-weighted step size (amu bohr)1/2

Log

arith

m o

f th

e av

erag

e er

ror

in th

eco

nser

vatio

n of

tota

l ene

rgy

(har

tree

) 5th order fit (3.00)

Rational fit (2.47)

2nd order fit (2.76)

Updating methods for Hessian-based trajectory integration

• Significant savings in CPU time if the Hessian can be updated for a few steps before being recalculated

• BFGS and BSP updating formulas used in geometry optimization are not suitable here

• Murtaugh-Sargent (Symmetric Rank 1) update is satisfactory

H new H old g Holdx g H oldx t

g H oldx tx

Relative CPU Time as a Function of the number of Hessian Updates

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 3 6 9Number of Hessian updates

Rel

ativ

e C

PU

usa

ge

C3N3H3

H2CO + CH3FNCCHO + CH3Cl

Dynamics with Atom-centered Basis Functions and Density Matrix Propagation (ADMP)

• Schlegel, Millam, Iyengar, Voth, Daniels, Scuseria, Frisch, JCP 114, 9758 (2001), JCP 115, 10291 (2001), JCP 117, 8694 (2002)

• Lagrangian in an orthonormal basis

L = ½ Tr(VTMV) + ½ Tr(( W )2) - E(R,P)- Tr[(P2-P)]

R, V, M – nuclear coordinates, velocities and massesP, W – density matrix and density matrix velocity in an orthonormal

basis (Löwdin or Cholesky) – fictitious electronic ‘mass’ matrix

E(R,P) – energy (electronic + VNN)Tr[(P2-P)] – constraint for idempotency and N-representability

Euler-Lagrange Equations of Motion

• equations of motion for the nuclei

M dR/dt2 = - E/R|P

• equations of motion for the density matrix

dP/dt2 = --½ [E/P|R+P+P-] -½

• integrate using velocity Verlet• iteratively solve for to impose the

idempotency constraints on P and W

• Basis functions for representing the electronic structure

• Method of orthonormalization• Calculation of the energy derivatives• Integration of the equations of motion• Satisfying the idempotency constraints• Mass weighting

Behind the Scenes: Choices and Challenges

• Car-Parrinello method (CP) PRL 55, 2471 (1985)

– expand in plane waves (appropriate for condensed phase)

– most integrals easy to calculate with fast Fourier transforms

– only Hellmann-Feynman terms required for gradients

• Atom-centered Density Matrix Propagation (ADMP)

– expand in atom centered functions (e.g. gaussians)

– use methods from standard molecular orbital calculations

– far fewer basis functions needed than plane waves

– fast routines for multi-centered integrals

– gradients require Hellmann-Feynman and Pulay terms

Basis functions for representing the electronic structure: plane waves vs atom centered functions

• molecular orbitals vs density matrix– arbitrary rotations among occupied molecular orbitals do

not change the energy or the density – density matrices become sparse for large systems and

calculations scale linearly• atomic orbitals vs orthonormal orbitals

– in an AO basis, changes in the density matrix reflect changes in bonding and changes in overlap

– in an AO basis, effect of changes in the overlap must be handled by the propagation

– in an orthonormal basis, changes in the overlap are handled separately and only changes in the bonding are handled by propagation

– equations of motion simpler in an orthonormal basis

Representing the electronic structure

Derivative of the energy wrt the density

• energy calculated with the purified density in an orthonormal basis (like CG-DMS method)

E = Tr[h P + ½ G(P) P] + VNN

P = 3 P2 – 2 P3 – McWeeny purified densityh, G(P) – one and two electron integral matrices

• derivative of the energy with respect to the density in an orthonormal basis

E/P|R = 3 F P + 3 P F – 2 F P2 – 2 P F P – 2 P2 F

F = h + G(P) – Fock matrix, Kohn-Sham matrix• derivative contains only occupied-virtual blocks (occupied-

occupied and virtual-virtual blocks are zero)• equations of motion satisfy idempotency condition to first

order

Derivative of the energy wrt the nuclei

E/R|P = Tr[dh/dR P +1/2 G(P)/R|P P] + dVNN/dR

where h, G and P are matrices in the orthonormal basis• transformation to the orthonormal basis

S’ = UT U, P = U P’ UT, h = U-T h’ U-1 , etc.

where S’, h’, etc. are matrices in the atomic orbital basis

E/R|P = Tr[dh’/dR P’ +1/2 G’(P’)/R|P’P’] + dVNN/dR

- Tr[F’ U-1dU/dR P’ + P’ dUT/dR U-T F’]

(obtained using UU-1 = I, UdU-1/dR = dU/dR U-1 and P2=P)

E/R|P = Tr[dh’/dR P’ +1/2 G’(P’)/R|P’ P’] + dVNN/dR

- Tr[F’ P’ dS’/dR P’] + Tr{[F, P] (Q dU/dRU-1-P U-TdUTdR)}

• for a converged SCF calculation, [F, P]=0

Derivatives of the transformation matrix

• for the Löwdin orthonormalization, U= S’1/2

U/R = si (i1/2 + j

1/2)-1 (siT S’/R sj) sj

T

where and s are the eigenvalues and eigenvectors of S’

• for the Cholesky basis, S’=UTU and U is upper triangular

(U/R U-1) = (U-TS’/R U-1) for = ½ (U-TS’/R U-1) for = 0 for

• for large, sparse systems, Cholesky is O(N) and the transformed F and P are sparse

Velocity Verlet step for the density

– symplectic integrators provide excellent energy conservation over long periods

Pi+1=Pi+Wit - -½ [E(Ri,Pi)/P|R+iPi+Pii-i] -½ t2/2

Wi+½=Wi - -½ [E(Ri,Pi)/P|R+iPi+Pii-i] - ½ t/2 = (Pi+1-Pi)/t

Wi+1=Wi+½ --½[E(Ri+1,Pi+1)/P|R+i+1Pi+1+Pi+1i+1-i+1]-½ t/2

– Lagrangian multipliers chosen so that the idempotency condition and its time derivative are satisfied, but must not affect the conservation of energy, etc.

Constraint for Idempotency: scalar mass P = P-P0 = W0t--1[E(R0,P0)/P|R+P0+P0-] t2/2

• P0 is idempotent, choose so that P2 = P• W0 and E(R0,P0)/P|R contain only occ-virt blocks but P0+P0-

contains only occ-occ and virt-virt blocks• closed form solution,

P0 P P0 = -(I-(I-4AAT)1/2)/2, Q0 P Q0 = (I-(I-4ATA)1/2)/2,

where A=P0{W0t - -1 [E(R0,P0)/P|R] t2/2}Q0 • iterative solution,

B (A+AT)2 + B2, P0 P P0 = -P0 B P0 , Q0 P Q0 = Q0 B Q0

• W must satisfy WP + P W = W

W = W* - PW*P - QW*Q

where W*=(P-P0)/t - -1 [E(R0,P0)/P|R] t/2

Mass-weighting

• in the course of a trajectory, core orbitals change more slowly than valence orbitals

• second derivatives wrt density matrix element involving core orbitals are much larger than for valence orbitals

• core functions can be assigned heavier fictitious masses so that their dynamics are similar to valence functions

• in initial tests, chosen to be a diagonal matrix = for valence orbitals

½ = ½ (2 (Fii + 2)1/2 +1) for core orbitals with Fii < -2 hartree

• more general choice may be desirable, depending on the elements and the basis set

Iterative solution for the Lagrangian multipliers for mass weighted case,

• i is chosen so that Pi+12 = Pi+1

(1) Pi+1 = Pi+Wit - -½ E(Ri,Pi)/P|R -½ t2/2

(2) P i+1 = 3 Pi+12 – 2 Pi+1

3

(3) i = ½ (P i+1- Pi+1) ½

(4) Pi+1= Pi+1+ -½ (Pi i Pi + Qi i Qi) -½

go to (2) if not converged

• i+1 is chosen so that Wi+1 Pi+1+ Pi+1 Wi+1 = Wi+1

(1) Wi+1= (Pi+1-Pi)/t - -½ E(Ri+1,Pi+1)/P|R -½ t/2(2) Wi+1= Wi+1- Pi+1 Wi+1 Pi+1 - Qi+1 Wi+1 Qi+1

(3) i+1= ½ (Wi+1- Wi+1) ½

(4) Wi+1= Wi+1+ -½ (Pi+1 i+1 Pi+1 + Qi+1 i+1 Qi+1) -½

go to (2) if not converged

Comparison of Resource Requirements• relative timings can be estimated from the times for computing the

Fock matrix, total energy, gradient and Hessian

– BO with Hessian based predictor-corrector

t = (t(energy) + t(Hessian)) / t

– BO with Hessian based predictor-corrector with 5 updates

t = (t(energy) + 1/6 t(Hessian) + 5/6 t(gradient)) / t

– BO with gradient based integrators with one gradient eval per t

t = (t(energy) + t(gradient)) / t

– ADMP with velocity Verlet integrator

t = (t(Fock) + t(gradient)) / t

• time step, t, chosen so that energy is conserved to ca 10-5 Hartree (t = ca 0.7 fs for Hessian methods, ca. 0.1 for gradient methods)

Relative timings for HF/6-31G(d) calculations on linear hydrocarbons

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 5.00 10.00 15.00 20.00

number of carbons

tim

ing

ra

tio BO-H/ADMP

BO-H5/ADMP

BO-G/ADMP

ADMP/ADMP

Relative timings for B3LYP/6-31G(d) calculations on linear hydrocarbons

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0.00 5.00 10.00 15.00 20.00

number of carbons

tim

ing

ra

tio BO-H/ADMP

BO-H5/ADMP

BO-G/ADMP

ADMP/ADMP

Chloride – Water Cluster

Method

Time step (fs)

Trajectory time (fs)

Conservation of energy (hartree)

PBE/3-21G* 0.06 120 0.0024 PBE/CEP-31G* 0.05 50 0.0001 PBE/CEP-31G* 0.10 45 0.0001 PBE/CEP-31G* 0.15 230 0.0002

(0.10 amu bohr2 (182 a.u.) fictitious ‘mass’ for the density)

Energy Conservation for Cl- (H2O)25

Comparison of the Fourier transform of the velocity-velocity autocorrelation function for

Cl- (H2O)25 at PBE/3-21G*

Fourier transform of the velocity-velocity autocorrelation function for Cl- (H2O)25 at

B3LYP/6-31G* (~1 ps)

(compare with 3620-3833 cm-1 for OH str in (H2O)2 and 3379 cm-1 in Cl--H2O)

Effect of Fictitious Mass on Vibrational Frequencies

• for CP calculations of ionic systems, effective mass of ion is the nuclear mass plus the fictitious mass of the electrons

• vibrational frequencies depend on the fictitious mass, but can be rescaled by a factor depending on the electron-nuclear coupling(a) (b)

Phonon density of states for (a) crystalline silicon and (b) crystalline MgO

P. Tangney and S. Scandolo, JCP 116, 14 (2002)

0 100 200 300 400

ADMP 0=0.4

ADMP 0=0.2

ADMP 0=0.1

B.O

R(N

a-C

l)

Time (femptosecond)

Effect of Fictitious Mass on Vibrational Frequencies

• for ADMP calculations, basis functions move with the nuclei• fictitious mass affects only the response of the density, not the

dynamics of the nuclei• vibrational frequencies do not depend on the fictitious mass

Vibrational motion of diatomic NaCl (fs)

Ab initio classical trajectory study of H2CO H2 + CO dissociation.

• Important test case, since studied intensively, both experimentally and theoretically.

• Excitation to S1 followed by internal conversion to S0, with sufficient energy to dissociate.

• Products are rotationally and vibrationally excited.• Born-Oppenheimer ab initio classical trajectory studies:

W. Chen, W. L. Hase, H. B. Schlegel, CPL 228, 436 (1994)

X. Li, J. M. Millam, H. B. Schlegel, JCP 113, 10062 (2000)

Effect of fictitious mass on energy conservation and adiabaticity

0 100 200 300 400 500 600 700

-0.02

-0.01

0.00

0.01

0.02

0 100 200 300 400 500

-0.02

-0.01

0.00

0.01

0.02

0 100 200 300 400

-0.02

-0.01

0.00

0.01

0.02

Ch

an

ge in

En

erg

y (

a.u

.)

(b)

Simulation Steps

(a)

(c)

valence = 0.40 amu bohr2

valence = 0.20 amu bohr2

valence = 0.10 amu bohr2

..... kinetic energy of the density

--- total energy

__ real energy (total – kinetic)

Effect of the Fictitious Mass on the Vibrational Distributions for CO and H2

Method

CO v=0

v=1

v=0

H2 v=1

v=2

v=3

Born-Oppenheimer 89 11 37 35 20 8

ADMP = 0.025 amu bohr2 87 13 36 35 20 9

= 0.050 amu bohr2 88 12 37 35 20 8

= 0.100 amu bohr2 87 13 37 35 21 7

Experiment 88 12 24 41 25 9

Effect of the Fictitious Mass on the Rotational Distributions for CO and H2

0 1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

Po

pu

lati

on

0.025 0.05 0.1

H2 Rotational Quantum Number

30 35 40 45 50 55 600.00

0.05

0.10

0.15

0.20

0.25

0.30

CO Rotational Quantum Number

Three Body Photofragmentation of Glyoxal Xiaosong Li, John M. Millam, H. B. Schlegel

JCP 114, 8 (2001), JCP 114, 8897 (2001), JCP 115 6907 (2001)

• Under collision free conditions, excitation to S1 followed by internal conversion to S0, with 63 kcal/mol excess energy.

• High pressure limit for rate constant: Ea = 55 kcal/mol.

• CO formed with Jmax = 42 and a broad distribution, but vibrationally cold

• H2 has significant population in v=1, but low J

O

OH

H

H2 + CO + CO

H2CO + CO

HCOH + CO

X

HCO + HCO

X

O

OH

H *h

Transition States for Glyoxal Unimolecular Dissociation

O

H

CC

H

O2.032

1.1001.403

1.151

<OCCO=90.3

<HCCH=9.6

H

CC

O

O

H

1.4831.2351.154

1.5381.401

<OCCH=108.8

C

O

C

OH

H

2.057

1.248

1.237

1.387

1.150

<OCCO=180

Barrier enthalpies (CBS-APNO) glyoxal H2 + 2 CO 59.2 kcal/mol H2CO + CO 54.4 kcal/mol HCOH + CO 59.7 kcal/mol 2 HCO 70.7 kcal/mol

C

O

C

O

H H

2.187

1.971 1.079

<OCCO=01.1661.200

CC

O

OH

H

2.142

1.9281.082

1.176

1.190

<OCCO=180

0 10.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

CO vibrational quantum number

Po

pu

lati

on

C.P. HF/3-21G B.O. HF/3-21G B.O. B3LYP/6-311G(d,p)

0 10.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

H2 vibrational quantum number

Vibrational energy distribution

ADMP HF/3-21G

Acknowledgments• Group members

– Xiaosong Li, Smriti Anand, Hrant Hratchian, Jie Li, Jason Cross, Stanley Smith, John Knox

• Collaborators– S. S. Iyengar, G. E. Scuseria, G. A. Voth, J. M. Millam, M. J. Frisch, W. L. Hase, Ö. Farkas, C. H. Winter, M. A. Robb, S. Shaik, T. Helgaker,

• Funding– National Science Foundation, DOE, Gaussian Inc.,

Wayne State University

Xiaosong Li, John Knox, Hrant Hratchian, Stan Smith, Smriti Anand, HBS, Jie Li