The ‘Multimode’ Approach to Challenging Problems in

Vibrational Spectroscopy

Joel M. Bowman, Stuart Carter, Xinchuan Huang

and Nicholas Handy

Important collaborators:

Bastiaan Braams and Anne McCoy

Support from the Office of Naval Research and the National Science Foundation

The challenges of ab initio spectroscopy

1. General and practical methods to solve the Schroedinger equation for nuclear motion

2. General and practical methods to obtain high quality ab initio-based potentials and dipole moment surfaces to use in step 1

Acetylene Hamiltonian in valence

coordinates

H

H

C C

rCH1 rCH2

rCC

M. Bramley and N. C. Handy, J. Chem. Phys. 98, 1378 (1993)

Kinetic Energy Operator

Coded in “RVIB4”Colwell, Carter and Handy

Jacobi Coordinates

r1

r2

φ

R

H

H C

C

Hamiltonian

r1

r2

φ

R

H

H C

C

€

H=TR+Tr1+Tr2+(J−j12)22μRR2 + j12

2μ1r12+j22

2μ2r22+V(θ2,θ1,φ,R,r1,r2)r j 12=

r j 1+r j 2

See CCP6 Library for a 4-atom code in Jacobi-Law,Tennyson, Hutson

Normal Coordinates Redux

In the 80s and 90s great progress was made in using

curvilinear

coordinates. Triatomics are a “solved problem”.

Extensions to

tetraatomics made in the 90s, but choice of coordinates

became

problem-specific, e.g., bond lengths and angles, umbrella,

Radau, Jacobi,

polyspherical, etc. Kinetic energy operators are coordinate-

specific

and can be quite complex. Not clear how this approach

would extend beyond 4 atoms or special case

pentatomics, e.g., CH4.

“RVIB3” (Carter-Handy), “RVIB4” (C-H-Colwell), “DVR3D”

(Tennyson), see CCP6, Carrington, Guo, Bowman, Leforstier,

Light, etc.

Bring back Normal Coordinates and

the Hamiltonians based on them

Start with the Watson Hamiltonian

€

ˆ H =1

2ˆ J α − ˆ π α( )μαβ

ˆ J β − ˆ π β( ) −1

2αβ

∑ ∂2

∂Qk2

−1

8μαα

α

∑k

∑ + V (Q)

€

Ψn1,n2,,...,nNVSCF (Q1,Q2,...QN ) = Π

i=1

Nφni

(i)(Qi ),

€

[Tl+ < Πi≠l

Nφni

(i ) | V +Tc | Πi≠l

Nφni

(i ) > −ε nl(l ) ]φnl

(l )(Ql ) = 0, l = 1,N

€

ΨKVCI = Ψn1,n2,,...,nN

VSCF + Cn1',n2 ',,...,nN 'K Ψn1',n2 ',,...,nN '

Virtuals

n1',n2 ',,...,nN '∑

Issues: Huge matrices and large dimensional integration

The Hamiltonian Matrix

€

H = T +V; HJ',J = J' T + V J

€

J(J ') = n1,n2 ,n3,K nN (n'1 ,n'2 ,n'3 ,K n'N )

Huge matrices and large dimensional integration

€

VJ',J = n'1 ,n'2 ,n'3 ,K n'N V(Q1,K ,QN ) n1,n2,n3,K nN

Numerical quadrature done in N dimensions?e.g., 6 dimensions for H3O+, 9 for H3O2

-, 15 for H5O2+

Matrix dimension is ca 10N

e.g., 106 for H3O+, 109 for H3O2-, 1015 for H5O2

+

MULTIMODE w/Stuart Carter at Emory 1996

€

V (Q1,Q2 ,...,QN ) = Vi(1) (Qi ) +

i∑ Vij

(2) (Qi ,Q j )ij∑ + Vijk

(3)

ijk∑ (Qi ,Q j ,Qk )

+ Vijkl(4)

ijkl∑ (Qi ,Q j ,Qk ,Ql ) + Vijklm

(5)

ijklm∑ (Qi ,Q j ,Qk ,Ql ,Qm ) + ...

The one-mode representation of the potential has V(1)

terms. i.e., “cuts” through the hyperspace of normal

coordinates with one coordinate varying. The two-mode

representation contains those terms plus the V(2) terms, etc.

n-mode representation of the potential

How this helps

€

VJ',J = n'1 ,n'2 ,n'3 ,K n'N V(Q1,K ,QN ) n1,n2,n3,K nN

=all triples

∑ n'i ,n' j ,n'k V(Qi ,Q j ,Qk ) ni ,n j ,nk δn'≠,n'i ,n' j ,n'k ,n≠ni ,n j ,nk

€

V 3MR (Q1,Q2 ,...,QN ) = Vi(1) (Qi ) +

i∑ Vij

(2) (Qi ,Q j )ij∑ + Vijk

(3)

ijk∑ (Qi ,Q j ,Qk )

So only have to do 3 dimensional quadratures and the matrix

is quite sparse, i.e., lots of zeros. Dimensionality of quadrature

space is nMR in general and matrix fills “slowly” as n increases

n-mode representation in MULTIMODE

For an n-mode representation of V in a problem with Nmodes there are N!/[n!(N-n)!] grids of dimension n.

Numerical integration is over these grids. Thus, the

dimensionality of quadratures is n < N; currently nmax = 6

Many matrix elements are zero.

The potential may be directly calculated on these quadrature grids or on sparser grids and interpolated.

Highly parallel procedure

H-Matrix Dimension

• Direct diagonalization feasible for dim ≈ 20,000

• Iterative methods used for dim up to 100,000

• Use symmetry to block diagonalize H

• Pick reference geometry to exploit max symmetry

MM should be exact for triatomics

0 0 0

ZPE 4637.97a 4637.97

b

Ka Kc

0 3 136.76 136.85 1 3 142.28 142.32 1 2 173.37 173.41 2 2 206.30 206.36 2 1 212.15 212.14 3 1 285.22 285.25 3 0 285.42 285.45

1 0 0

Ka Kc

0 3 1731.92 1732.77 1 3 1739.52 1739.68 1 2 1772.44 1772.63 2 2 1813.82 1813.88 2 1 1819.36 1818.60 3 1 1907.48 1907.38 3 0 1907.64 1907.52

2 0 0

Ka Kc

0 3 3289.23 3290.64 1 3 3299.99 3300.25 1 2 3334.62 3334.93 2 2 3387.68 3387.74 2 1 3392.74 3391.43 3 1 3500.51 3500.30 3 0 3500.64 3500.39

0 1 0

Ka Kc

0 3 3791.36 3791.45 1 3 3796.53 3796.59 1 2 3827.37 3827.44 2 2 3858.86 3858.98 2 1 3864.76 3864.83 3 1 3935.20 3935.37 3 0 3935.34 3935.57

aPartridge and Schwenke (1997)bMM - Carter and Bowman (1998).

H2O J = 3

Vibrational energies of CH4 (J=0)a

State H.O. 2MR 3MR 4MR Radaub

ZPE

4(F2)

2(E)

1(A1)

24

2 + 4

3(F2)

22

9835.0

1344.0

1570.8

3034.7

2688.0

2914.8

3153.9

3141.6

9693.1

1311.7

1531.1

2925.7

2626.1

2881.6

3004.3

3067.3

9707.4

1312.9

1534.4

2948.3

2621.6

2831.5

3053.7

3067.2

9707.2

1313.3

1534.5

2949.4

2623.9

2836.4

3053.1

3067.3

------

1314.1

1534.0

2955.8

2627.2

2838.1

3056.5

3069.0

a Carter and Bowman, J. Chem. Phys. (1999) using Taylor-Lee-Martin ab initio force field. bYu, J. Chem. Phys. (2003)

Recent benchmark calculations by Carrington using Radau

What about V?

• H2CO H2+CO, H+HCO

JPC (2004)

• (OH-)H2O JACS (2004)

• H5O2+ JCP (2004)

• (H2O)2 (done)

• O(3P)+C3H3 JTCC (2005)

• CH5+ JCP (2003), in progress

• C(3P, 1D)+C2H2 in progress

• H5+ JCP (2005)

• CH3OH (done)

MP2,CCSD(T), MRCI ab initio calculations done (MOLPRO) on grids or using“direct dynamics”. Fits in inter-nuclear distances enforcing permutationalsymmetry. ca 50 000 ab initio energiesfeasible on our 100 cpu cluster - DURIP

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

“Multinode”

Floppy hydrated systems

• H3O+ (Huang, Carter, … )

• H3O2- (Braams, Huang, Carter, McCoy*,…)

• H5O2+ (Braams, Huang, Carter, McCoy*,…)

*DMC calculations, “easy” for the ground state, but requirethe real McCoy (expertise) for excited states.

H3O+

Ammonia like inversion motion - a challenge for MM

Where to put the reference geometry/what normalmodes to use?

We picked the saddle point - D3h and had to use alarge grid in the imaginary frequency mode to spanthe two minima.

H3O+

2000

1500

250

250

1000

1000

750

750

500500

-50 -25 0 25 50

20

10

0

-10

Q1

Q2

Ab initio-based (CCSD(T)) potential energy surface in two modes - contour values in cm-1.

Using a form for the potential developed by Leonard, Carter and Handy for NH3

Comparison of calculated and experimental vibrationalsplittings (cm-1) in H3O+ and D3O+ using MM and RVIB4

THEORY EXP

ZPE 46.0 55.3

354.0 373.

560.0 NA

4 57. 0 68. 0

37. 0 46.

3 3. 0 38. 7

H3O+

THEORY EXP

ZPE 12.0 15.3 74 .0 9.4

363.0 NA

4 5.0 NA

.0 NA

3 7.0 0.0

D3O+

Halonen and co-workers have developed an even more accurate PES and getbetter agreement with experiment.

Exp - Oka, Sears, Saykally

Theory - X. Huang, S. Carter and J. M. Bowman, J. Chem. Phys. 118, 5431 (2003).

[J >> 0 Chakrobarty, Truhlar, Bowman, Carter (2004)]

David Nesbitt, private communication, JCP (2005)

H3O2- aka (OH-)H2O

• Experiments by Johnson and co-workers

• Full dim PES (Huang, Braams, Bowman)

• MM-Reaction Path version* (Huang, Carter,JMB)

• DMC calculations (McCoy)

*Carter and Handy (2002), Miller, Handy, Adams (1988), Hougen, Bunker, Johns (1970)

OH-[H2O] Exp and Previous Theory

• H3O2- H2O + OH-

• Structure and HO:Xantheas (1995)• Estimated binding energy:

8,300 – 9,900 cm-1 • H-atom transfer barrier:

50 – 100 cm-1 Klopper (2002)

• H delocalization :300 K, Tuckerman, Parrinello (1997)

• Ar-Predissociation IR spectrum:Johnson et al. (2002)

Potential minimum

C1 symmetry

H-transfer barrier and HO in H3O2-

Barrier height = 74 cm-1 in good agreement w/Sampson & Klopper

Mod

e

1 2 3 4 5 6 7 8 9

SP 621i 213 572 576 631 152

5

161

8

381

4

381

5

MIN 191 305 476 596 135

3

155

4

170

2

378

4

383

5

Results at MIN are closeto Xantheas’ 1995 MP2calculations and in rough“agreement” with exp.But HO results at SP disagreewith experiment. IR intensitiesalso disagree.

H3O2- PES

63675 CCSD(T)/aug-cc-pVTZ energiesUsing MOLPRO 2001.

Use MM-grids to generate configs

Fit (Braams) using a basisthat is symmetric wrt interchangeof like atoms - use all internuclear distances as basic variables; Morse, SPF, etc.

2021 coefficients (incomplete 7th order terms)

Standard Deviation : 16 cm-1

Average fitting error : 9 cm-1

Range (cm-1) No. of Points RMS Error (cm-1)

0 – 5000 10050 5.8

0 – 10000 25835 9.0

0 – 15000 44025 11.5

0 – 25000 59545 14.2

0 – 50000 63317 15.4

0 10000 20000 30000 40000 500000

1000

2000

3000

4000

Data Count / pts

E (cm-1)

Energy Distribution of ab initio points used in H3O2- PES-1

0 60 120 180 240 300 3600

100

200

300

400

500

Torsion Angle / degree

Torsional potential (cm-1) along C2-path

(Like H2O2 with a bridging H-atom-w/smaller torsional barriers)

ZPE is muchgreater than thebarrier separatingthe two equivalentminima.

Harmonic Frequencies (cm-1) along C2-path

H3O2-

D3O2-

OO-stretch/wag“avoided crossing”

OO-stretch/wag“avoided crossing”absent for D3O2

-.

Argon Predissociation Spectroscopy of the OH-‚H2O and Cl-‚H2O

Complexes in the 1000-1900 cm-1 Region: Intramolecular Bending

Transitions and the Search for the Shared-Proton Fundamental

in the Hydroxide Monohydrate

Eric G. Diken, Jeffrey M. Headrick, Joseph R. Roscioli, Joseph C. Bopp, and

Mark A. Johnson*Anne B. McCoy* Xinchuan Huang, Stuart Carter, and Joel M.

Bowman*

J. Phys. Chem. A 2005, 109, 571-575

Bridging H-atom stretch

The double HO approx givesvery small intensity for this mode at the min. Buta much larger intensity at thesaddle point. (But an imaginaryfrequency.)

H5O2+ - The Current Limit

H5O2+ - The Current Limit

• Potential and dipole moment in full dimensionality • Vibrational calculations -MM and DMC-McCoy• Experiments (Johnson group)

?

JCP (in press)

Erratum: JMB Talk on Tuesday

(Bowman instantly had second thoughts but itwas too late…)

Bowman stated that no experimental spectra hadbeen reported for H5

+