Using Valence Bond Theory to Model (Bio)Chemical Reactivitycompchemkitchen.org/wp-content/uploads/2016/06/cck2_fd.pdf · CCK-2, June 14th 2016 1 Fernanda Duarte Using Valence Bond

CCK-2, June 14th 2016

1

Fernanda Duarte

Using Valence Bond Theory to Model

(Bio)Chemical Reactivity

Comp Chem Kitchen

Department of Chemistry

2

Motivation

Explicit incorporation of environment effects

Most FFs (AMBER, OPLSAA, CHARMM) assume the same atomic connectivity throughout simulation. No Chemistry!!!

Molecular Dynamics

Quantum MechanicsElectrons, breaking/forming bond processes, exited state reactions

Computationally demanding (N4-N7): DFT

Aims to combine the best of each world...

Sampling/computational cost still an issue

QM/MM

E = Ψ H QM + HelQM /MM Ψ +Evan

QM /MM +EMM

Ψ = c1ψ1 + c2ψ2 +....ciψi

3

Valence Bond (VB) Approach

Valence bond ApproachVB looks for probability of finding a given molecular state

RS TS PS

‡

H =

V11 V12 V1 j! " !Vi1 Vi2 Vij

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Each matrix element is a function of molecular geometry,q. Good approximations for V11 and V22 are available frommolecular mechanics. However, much less is known aboutthe functional form of the interaction matrix element, V12.In Warshel and Weiss’s approach,2 the interaction matrix

element V12 is chosen to reproduce the barrier height(obtained from experiments or calculations). For cases wheregreater accuracy is required, it is also desirable to match theposition and vibrational frequencies of the TS in addition tothe barrier height. The Chang-Miller approach3 describesthe interaction matrix element by a generalized Gaussianpositioned at or near the transition state

where qTS is the transition-state geometry. The coefficientsare chosen so that the energy, gradient, and second deriva-tives of the EVB surface match ab initio calculations at theTS. Following Chang-Miller’s notation,3 this yields simple,closed-form equations for parameters A, B (a vector), andC (a matrix).

The original version of the Chang-Miller method runsinto difficulties when C has one or more negative eigenval-ues.12,15,18 In these cases, the form of V122 in eq 5 divergesfor large ∆q values. The simplest solution to this problemswitches the interaction matrix element to zero in regionswhere the unmodified V122 is negative or divergent.15 Anotherapproach is to include suitable cubic and quartic terms inthe Gaussian to control asymptotic behavior.12In the present article, an alternative form for V122 is pro-

posed. Instead of using a generalized Gaussian as in eq 5, aquadratic polynomial times a spherical Gaussian is employed.

Fitting to the energy, gradient, and Hessian at the transitionstate yields the same formulas for A and B as those in the

Chang-Miller case; the expression for C is slightly different.

The exponent R is chosen to be small enough so that thePES is smooth but not so small that the reactant and productenergies are affected significantly. One approach is to chooseR to give a good fit for the energies along the reaction path.The form of V122 in eq 7 can also be viewed as expandingV122 as a linear combination of s-, p-, and d-type Gaussians.

Because the coefficients in eq 7 are linear, the procedurecan be readily generalized to include Gaussians at multiplecenters, qK. For example, one could choose to place theGaussian centers at the TS, reactant minimum, productminimum, and a few points along the reaction path to eitherside of the transition state. The generalized form of V122 canbe written as

where NDim is 3 times the number of atoms for a Cartesiancoordinate system or the number of coordinates if internalor redundant-internal coordinates are utilized. The Gaussianexponents are chosen such that the fit is sufficiently smoothfor energies along the reaction path and V122 is acceptablysmall at the reactants and products, if these are not alreadyincluded in qK. In the simplest approach, the exponents areall equal; alternatively, if suitable criteria exist, they maybe different for different centers, or even for differentdirections. The BijK coefficients are obtained by fitting toV122 and its first and second derivatives at a number of points,qL, which can conveniently be the same as qK.

If the number of Gaussian centers is equal to the number ofpoints (i.e., if the number of coefficients is equal to thenumber of energy values, first derivatives, and secondderivatives), this is simply the solution of a set of linearequations.

V11 ) ⟨ψ1|H|ψ1⟩, V12 ) V21 ) ⟨ψ1|H|ψ2⟩,V22 ) ⟨ψ2|H|ψ2⟩ (3)

V ) 1/2(V11 + V22) - ![1/2(V11 - V22)]2 + V12

2 (4)

V122(q) ) A exp[BT‚∆q - 1/2∆q

T‚C‚∆q], ∆q ) q - qTS(5)

A ) [V11(qTS) - V(qTS)][V22(qTS) - V(qTS)] (6a)

B )D1

[V11(qTS) - V(qTS)]+

D2[V22(qTS) - V(qTS)]

and

Dn )∂Vnn(q)

∂q |q)qTS-∂V(q)∂q |q)qTS (6b)

C )D1D1

T

[V11(qTS) - V(qTS)]2 +

D2D2T

[V22(qTS) - V(qTS)]2 -

K1V11(qTS) - V(qTS)

-K2

V22(qTS) - V(qTS)and

Kn )∂2Vnn(q)

∂q2|q)qTS

-∂2V(q)∂q2

|q)qTS

(6c)

V122(q) ) A[1 + BT‚∆q + 1/2∆q

T‚(C + RI)‚∆q]exp[-1/2R|∆q|2] (7)

C )D1D2

T + D2D1T

A +K1

V11(qTS) - V(qTS)+

K2V22(qTS) - V(qTS)

(8)

g(q,qK,0,0,R) ) exp[-1/2R|q - qK|2]

g(q,qK,i,0,R) ) (q - qK)i exp[-1/2R|q - qK|

2]

g(q,qK,i,j,R) ) (q - qK)i(q - qK)j exp[-1/2R|q - qK|

2] (9)

V122(q) )∑

K∑igjg0

NDim

BijKg(q,qK,i,j,R) (10)

V122(qL) )∑

K∑igjg0

NDim

BijKg(qL,qK,i,j,R)

∂V122(q)

∂q|q)qL

)∑K∑igjg0

NDim

BijK∂g(q,qK,i,j,R)

∂q|q)qL

∂2V122(q)

∂q2|q)qL

)∑K∑igjg0

NDim

BijK∂2g(q,qK,i,j,R)

∂q2|q)qL

(11)

DB ) F (12)

906 J. Chem. Theory Comput., Vol. 2, No. 4, 2006 Schlegel and Sonnenberg

4

Empirical Valence Bond (EVB)

Empirical Valence-Bond Models for Reactive PotentialEnergy Surfaces Using Distributed Gaussians

H. Bernhard Schlegel* and Jason L. SonnenbergDepartment of Chemistry, Wayne State UniVersity, Detroit, Michigan 48202

Received March 3, 2006

Abstract: A new method for constructing empirical valence bond potential energy surfaces forreactions is presented. Building on the generalized Gaussian approach of Chang-Miller, V122(q)is represented by a Gaussian times a polynomial at the transition state and generalized to handleany number of data points on the potential energy surface. The method is applied to two modelsurfaces and the HCN isomerization reaction. The applications demonstrate that the presentmethod overcomes the divergence problems encountered in some other approaches. The useof Cartesian versus internal or redundant internal coordinates is discussed.

IntroductionIn the empirical valence bond (EVB) approach, the potentialenergy surface (PES) for a reaction in solution is modeledas an interaction between a reactant and a product PES.1 Theinteraction between surfaces results in an avoided crossingand yields a smooth function describing the reaction on theground-state potential energy surface. Good empirical ap-proximations for the noninteracting potential energy surfacesof reactants and products are available from molecularmechanics methods. To obtain a reliable model of a PESfor a reaction, a suitable form of the interaction matrixelement or resonance integral, V12(q), is needed.For a two-state system, the interaction between reactant

and product surfaces is taken as a modified Morse functionin Warshel and Weiss’ original multistate EVB method.2 Thefunction is adjusted to reproduce barrier heights gleaned fromexperiments or high-level ab initio calculations, but the formof the surface is not flexible enough to fit frequencies at thetransition state (TS). Chang and Miller represented the squareof the resonance integral, V122, with a generalized Gaussian.3The exponents of the Gaussian are chosen to fit the structureand vibrational frequencies of the TS from electronicstructure calculations. This form of the EVB surface issufficiently accurate for molecular dynamics.4-12 The Chang-Miller model has also been applied by Jensen13,14 andAnglada et al.15 to transition-state optimizations.More elaborate functions of the interaction matrix elements

were used in the molecular mechanics/valence bond modeldeveloped by Bernardi et al. for exploring photochemicalreaction potential energy surfaces.16 Minichino and Voth

generalized the Chang-Miller method3 for N-state systemsand provided a scheme to correct gas-phase ab initio datafor solutions.17 Truhlar and co-workers employed a general-ized EVB approach by using distance-weighted interpolantsto model the interaction matrix elements in their multicon-figuration molecular mechanics method.18-20

The simplicity of the Chang-Miller resonance integralformulation is appealing, but certain difficulties must beovercome to provide the greater flexibility required to modelmore complex chemical reactions using molecular dynamics.The present article explores two possibilities for improvingthe representation of the interaction matrix elements. Inparticular, the generalized Gaussian utilized in the Chang-Miller approach is replaced with a quadratic polynomialtimes a spherical Gaussian. This avoids the well-knownproblem caused by negative exponents that may arise inpractice.12,15,18 Second, to improve the accuracy of the fit, alinear combination of Gaussians times quadratic polynomialsplaced at suitable locations on the potential energy surfaceis employed.

Model DescriptionThe EVB model describes a reactive PES in terms of a linearcombination of reactant and product wave functions. Thecoefficients are obtained by solving a simple 2 × 2Hamiltonian for the lowest energy.

* Corresponding author. E-mail: [email protected].

Ψ ) c1ψ1 + c2ψ2 (1)

H ) [V11 V12V21 V22 ] (2)

905J. Chem. Theory Comput. 2006, 2, 905-911

10.1021/ct600084p CCC: $33.50 © 2006 American Chemical SocietyPublished on Web 05/09/2006

1)












Ψ ) c1ψ1 + c2ψ2 (1)

H ) [V11 V12V21 V22 ] (2)














V11 ) ⟨ψ1|H|ψ1⟩, V12 ) V21 ) ⟨ψ1|H|ψ2⟩,V22 ) ⟨ψ2|H|ψ2⟩ (3)

V ) 1/2(V11 + V22) - ![1/2(V11 - V22)]2 + V12

2 (4)

V122(q) ) A exp[BT‚∆q - 1/2∆q

T‚C‚∆q], ∆q ) q - qTS(5)


B )D1



and

Dn )∂Vnn(q)


C )D1D1

T

[V11(qTS) - V(qTS)]2 +

D2D2T

[V22(qTS) - V(qTS)]2 -

K1V11(qTS) - V(qTS)

-K2


Kn )∂2Vnn(q)

∂q2|q)qTS

-∂2V(q)∂q2

|q)qTS

(6c)

V122(q) ) A[1 + BT‚∆q + 1/2∆q

T‚(C + RI)‚∆q]exp[-1/2R|∆q|2] (7)

C )D1D2

T + D2D1T

A +K1

V11(qTS) - V(qTS)+

K2V22(qTS) - V(qTS)

(8)

g(q,qK,0,0,R) ) exp[-1/2R|q - qK|2]


2]


2] (9)

V122(q) )∑

K∑igjg0

NDim


V122(qL) )∑

K∑igjg0

NDim

BijKg(qL,qK,i,j,R)

∂V122(q)

∂q|q)qL

)∑K∑igjg0

NDim


∂q|q)qL

∂2V122(q)

∂q2|q)qL

)∑K∑igjg0

NDim


∂q2|q)qL

(11)

DB ) F (12)













V11 ) ⟨ψ1|H|ψ1⟩, V12 ) V21 ) ⟨ψ1|H|ψ2⟩,V22 ) ⟨ψ2|H|ψ2⟩ (3)

V ) 1/2(V11 + V22) - ![1/2(V11 - V22)]2 + V12

2 (4)

V122(q) ) A exp[BT‚∆q - 1/2∆q

T‚C‚∆q], ∆q ) q - qTS(5)


B )D1



and

Dn )∂Vnn(q)


C )D1D1

T

[V11(qTS) - V(qTS)]2 +

D2D2T

[V22(qTS) - V(qTS)]2 -

K1V11(qTS) - V(qTS)

-K2


Kn )∂2Vnn(q)

∂q2|q)qTS

-∂2V(q)∂q2

|q)qTS

(6c)

V122(q) ) A[1 + BT‚∆q + 1/2∆q

T‚(C + RI)‚∆q]exp[-1/2R|∆q|2] (7)

C )D1D2

T + D2D1T

A +K1

V11(qTS) - V(qTS)+

K2V22(qTS) - V(qTS)

(8)

g(q,qK,0,0,R) ) exp[-1/2R|q - qK|2]


2]


2] (9)

V122(q) )∑

K∑igjg0

NDim


V122(qL) )∑

K∑igjg0

NDim

BijKg(qL,qK,i,j,R)

∂V122(q)

∂q|q)qL

)∑K∑igjg0

NDim


∂q|q)qL

∂2V122(q)

∂q2|q)qL

)∑K∑igjg0

NDim


∂q2|q)qL

(11)

DB ) F (12)


5

Diagonal Elements (Vii)/Coupling Element (Vij)

( )( )2

stretch 01 expeV D R Rβ⎡ ⎤= − −⎣ ⎦( )2stretch 012 RV k R R= −

( )2bend 012

V kθ θ θ= − ( )torsion 1 cos ; 1nV V s n sφ= + = ±⎡ ⎤⎣ ⎦

Bending and Torsional potentials:

Bond Stretch

Non-bonding potentials:

Lennard-Jones 12 6ij ij

ij ij

A CV

R R= −

2

Coulombi j

ij

q q eV

R=

12

6

; 4

; 4

ij i j i i i

ij i j i i i

A A A A

C CC C

ε σ

ε σ

= =

= =












V11 ) ⟨ψ1|H|ψ1⟩, V12 ) V21 ) ⟨ψ1|H|ψ2⟩,V22 ) ⟨ψ2|H|ψ2⟩ (3)

V ) 1/2(V11 + V22) - ![1/2(V11 - V22)]2 + V12

2 (4)

V122(q) ) A exp[BT‚∆q - 1/2∆q

T‚C‚∆q], ∆q ) q - qTS(5)


B )D1



and

Dn )∂Vnn(q)


C )D1D1

T

[V11(qTS) - V(qTS)]2 +

D2D2T

[V22(qTS) - V(qTS)]2 -

K1V11(qTS) - V(qTS)

-K2


Kn )∂2Vnn(q)

∂q2|q)qTS

-∂2V(q)∂q2

|q)qTS

(6c)

V122(q) ) A[1 + BT‚∆q + 1/2∆q

T‚(C + RI)‚∆q]exp[-1/2R|∆q|2] (7)

C )D1D2

T + D2D1T

A +K1

V11(qTS) - V(qTS)+

K2V22(qTS) - V(qTS)

(8)

g(q,qK,0,0,R) ) exp[-1/2R|q - qK|2]


2]


2] (9)

V122(q) )∑

K∑igjg0

NDim


V122(qL) )∑

K∑igjg0

NDim

BijKg(qL,qK,i,j,R)

∂V122(q)

∂q|q)qL

)∑K∑igjg0

NDim


∂q|q)qL

∂2V122(q)

∂q2|q)qL

)∑K∑igjg0

NDim


∂q2|q)qL

(11)

DB ) F (12)













V11 ) ⟨ψ1|H|ψ1⟩, V12 ) V21 ) ⟨ψ1|H|ψ2⟩,V22 ) ⟨ψ2|H|ψ2⟩ (3)

V ) 1/2(V11 + V22) - ![1/2(V11 - V22)]2 + V12

2 (4)

V122(q) ) A exp[BT‚∆q - 1/2∆q

T‚C‚∆q], ∆q ) q - qTS(5)


B )D1



and

Dn )∂Vnn(q)


C )D1D1

T

[V11(qTS) - V(qTS)]2 +

D2D2T

[V22(qTS) - V(qTS)]2 -

K1V11(qTS) - V(qTS)

-K2


Kn )∂2Vnn(q)

∂q2|q)qTS

-∂2V(q)∂q2

|q)qTS

(6c)

V122(q) ) A[1 + BT‚∆q + 1/2∆q

T‚(C + RI)‚∆q]exp[-1/2R|∆q|2] (7)

C )D1D2

T + D2D1T

A +K1

V11(qTS) - V(qTS)+

K2V22(qTS) - V(qTS)

(8)

g(q,qK,0,0,R) ) exp[-1/2R|q - qK|2]


2]


2] (9)

V122(q) )∑

K∑igjg0

NDim


V122(qL) )∑

K∑igjg0

NDim

BijKg(qL,qK,i,j,R)

∂V122(q)

∂q|q)qL

)∑K∑igjg0

NDim


∂q|q)qL

∂2V122(q)

∂q2|q)qL

)∑K∑igjg0

NDim


∂q2|q)qL

(11)

DB ) F (12)


Vsoft = Ci · Cj · e(-ai ·aj ·ri, j )

Lennard-Jones 12 6ij ij

ij ij

A CV

R R= −

2

Coulombi j

ij

q q eV

R=

12

6

; 4

; 4

ij i j i i i

ij i j i i i

A A A A

C CC C

ε σ

ε σ

= =

= =












V11 ) ⟨ψ1|H|ψ1⟩, V12 ) V21 ) ⟨ψ1|H|ψ2⟩,V22 ) ⟨ψ2|H|ψ2⟩ (3)

V ) 1/2(V11 + V22) - ![1/2(V11 - V22)]2 + V12

2 (4)

V122(q) ) A exp[BT‚∆q - 1/2∆q

T‚C‚∆q], ∆q ) q - qTS(5)


B )D1



and

Dn )∂Vnn(q)


C )D1D1

T

[V11(qTS) - V(qTS)]2 +

D2D2T

[V22(qTS) - V(qTS)]2 -

K1V11(qTS) - V(qTS)

-K2


Kn )∂2Vnn(q)

∂q2|q)qTS

-∂2V(q)∂q2

|q)qTS

(6c)

V122(q) ) A[1 + BT‚∆q + 1/2∆q

T‚(C + RI)‚∆q]exp[-1/2R|∆q|2] (7)

C )D1D2

T + D2D1T

A +K1

V11(qTS) - V(qTS)+

K2V22(qTS) - V(qTS)

(8)

g(q,qK,0,0,R) ) exp[-1/2R|q - qK|2]


2]


2] (9)

V122(q) )∑

K∑igjg0

NDim


V122(qL) )∑

K∑igjg0

NDim

BijKg(qL,qK,i,j,R)

∂V122(q)

∂q|q)qL

)∑K∑igjg0

NDim


∂q|q)qL

∂2V122(q)

∂q2|q)qL

)∑K∑igjg0

NDim


∂q2|q)qL

(11)

DB ) F (12)


V12 = Aexp{−a(r1 − r10 )2}

V12 = Aexp{−(a(r1 − r10 )2 + 2b(r1 − r1

0 )(r2 − r20 )+ c(r2 − r1

0 )2 )}

Warshel, 1980

Glowacki, 2011

V12 = Aexp{BTΔq− 0.5ΔqT ⋅C ⋅ Δq}, Δq = q−qTS

Chang-Miller, 1990

Position and frequencies of the TS

Barrier Height

6

Potential Energy Surface

TS

PSRS

2) Fit V12 to reproduce experimental or (QM) data

Diagonalize V adiabatic states. The minimal value is the ground state.

3)












V11 ) ⟨ψ1|H|ψ1⟩, V12 ) V21 ) ⟨ψ1|H|ψ2⟩,V22 ) ⟨ψ2|H|ψ2⟩ (3)

V ) 1/2(V11 + V22) - ![1/2(V11 - V22)]2 + V12

2 (4)

V122(q) ) A exp[BT‚∆q - 1/2∆q

T‚C‚∆q], ∆q ) q - qTS(5)


B )D1



and

Dn )∂Vnn(q)


C )D1D1

T

[V11(qTS) - V(qTS)]2 +

D2D2T

[V22(qTS) - V(qTS)]2 -

K1V11(qTS) - V(qTS)

-K2


Kn )∂2Vnn(q)

∂q2|q)qTS

-∂2V(q)∂q2

|q)qTS

(6c)

V122(q) ) A[1 + BT‚∆q + 1/2∆q

T‚(C + RI)‚∆q]exp[-1/2R|∆q|2] (7)

C )D1D2

T + D2D1T

A +K1

V11(qTS) - V(qTS)+

K2V22(qTS) - V(qTS)

(8)

g(q,qK,0,0,R) ) exp[-1/2R|q - qK|2]


2]


2] (9)

V122(q) )∑

K∑igjg0

NDim


V122(qL) )∑

K∑igjg0

NDim

BijKg(qL,qK,i,j,R)

∂V122(q)

∂q|q)qL

)∑K∑igjg0

NDim


∂q|q)qL

∂2V122(q)

∂q2|q)qL

)∑K∑igjg0

NDim


∂q2|q)qL

(11)

DB ) F (12)













Ψ ) c1ψ1 + c2ψ2 (1)

H ) [V11 V12V21 V22 ] (2)



1)












Ψ ) c1ψ1 + c2ψ2 (1)

H ) [V11 V12V21 V22 ] (2)



7

I

X

O

RO

O

NO2

O

OH

FGl57H

X

O

R

OHO

NO2

H N

H

H

K337

HN

H

K337

O

OH

FGl57H

X=P,S

R=O-, OCH2CH3, Ph

Empirical Valence Bond (EVB): Enzyme Catalysis

∆G‡

[kca

l ·m

ol -1

]

0

5

10

15

20

WT Q13A N78A Y105A T107A H218 K337

B.

H218A K337AT107AY105AN78AQ13AWT

Barrozo/Duarte et al. J. Am. Chem. Soc. 2015, 137, 9061

Phopshate Hydrolysis (sN2 type reaction)

8

Different Flavours of VB

Program Capabilities Website Comments Reference

MOLARIS-XG EVB, FEP, AC http://laetro.usc.edu/software.html

Available for purchase from the USC

Warshel and coworkers

Q EVB, FEP, LIEhttp://xray.bmc.uu.se/~aqwww/

q/Free for academic use. Available upon request

Åqvist and coworkers

V2000VBSCF, BOVB, VBCI,SCVB, CASVB GVB http://www.scinetec.com Integrated into GAMESS

McWeeny and coworkers

TURTLE VBSCFhttp://tc5.chem.uu.nl/ATMOL/

turtle/turtle_main.html Integrated into GAMESS-UK van Lenthe et al.

VM/MM VB/MMDE-VB/MM

Available upon request [email protected].

Interface program that communicates between XMVB

and MOLARISShurki and coworkers

MS-EVB MS-EVB In house implementation in LAMMPS MD package

Voth and coworkers

AMBERDistributed Gaussian

EVB http://ambermd.org/AMBER license is required for

GPU versionCase and coworkers

XMVBVBSCF, BOVB, VBCI,

VBPT2, DFVB, VBPCM, VBEFP, VBEFP/PCM

http://ftcc.xmu.edu.cn/xmvb/index.html Integrated into GAMESS Wu and coworkers

http://tc5.chem.uu.nl/ATMOL/turtle/turtle_main.html

mailto:[email protected]

Documents

Using Valence Bond Theory to Model (Bio)Chemical Reactivitycompchemkitchen.org/wp-content/uploads/2016/06/cck2_fd.pdf · CCK-2, June 14th 2016 1 Fernanda Duarte Using Valence Bond