Using Valence Bond Theory to Model (Bio)Chemical Reactivity

Home , Tinker (software)

Master Course for Theoretical Chemistry and Computational Modelling (TCCM)

Fernanda Duarte Department of Chemistry, Oxford

September 7th 2016

1 General Outline of Lecture

• Motivation & History of VB

• Basic Concepts

• ab initio VB theory

• Multiscale VB simulations

• Empirical VB theory Empirical Valence Bond (EVB)

Semi-empirical QM/MM approach that uses fully classical description of key states within VB framework to describe chemical reactivity

H X C Y

H H

1 4 4 4-e/3-orbital/centre VB system g4,3 = 0 4 3 2 ✓ ◆✓ ◆

3 Empirical Valence Bond (EVB) H X C Y

H H

Parameter Abbreviation Value

Halogen (X) atom IPX 250

EAX 70

Methyl (CH3) group IPCH3 230

EACH3 ≈0

C-X covalent bond DC-X 60 4.0 Å X-X bond DX-X 0 + -- Charge distribution VQQ(+ --) -60 - + - Charge distribution VQQ(-+-) -180 Approximate Valence-Bond State Energies E(1) = reference state energy = 0 E(2) = reference state energy = 0

E(3) = VQQ(-+-) + IPCH3, - EAX + DC-X= +40

E(4) = VQQ(+--) + IPX - EACH3 + DC-X= +250

E(5) = VQQ(+--) + IPX - EACH3 + DC-X = +250

E(6) = IPX - EACH3 + DC-X - DX-X = +310

Warshel & Weiss, J. Am. Chem. Soc. 1980, 102, 6218 4 Empirical Valence Bond (EVB)

300 State 6 (1) State 4 State 5 X + H3C Y

(2) X CH + Y 200 3

(3) X + CH3 + Y

100 (4) X + CH3 + Y

State 3 (5) X + H3C + Y 0 State 1 State 2 (6) X + CH3 + Y E(kcal/mol)

5 Warshel & Weiss J. Am. Chem. Soc., 1980, 102, 6218 Empirical Valence Bond (EVB)

300 State 6 (1) X + H3C Y State 4 State 5 (2) X CH3 + Y

200 (3) X + CH3 + Y

(4) X + + CH3 + Y

100 (5) X + H3C + Y

State 3 (6) X + CH3 + Y 0 State 1 State 2 E(kcal/mol) ψ1 = α 1Φ1 + β 1Φ3

ψ2 = α 2 Φ1 + β 2 Φ3

6 Warshel & Weiss J. Am. Chem. Soc., 1980, 102, 6218 Empirical Valence Bond (EVB)

Two-State Problem

ψ1 = α 1Φ1 + β 1Φ3 (Hij "Sij)ci =0 ψ = α Φ + β Φ i=1 2 2 1 2 3 X

H " H "S 0= 11 12 12 H "S H " 21 21 22

7 Empirical Valence Bond (EVB)

Two-State Problem

H " H "S 0= 11 12 12 H "S H " 21 21 22 zero overlap between states!

Diagonal Elements Off-diagonals Elements H11 = 1 H 1 H = H h | | i 12 h 1| | 2i H = H 22 h 2| | 2i

8 Empirical Valence Bond (EVB)

Diagonal Elements: VB states are described by parabolic energy surfaces, using a classical FF description. H = H 11 h 1| | 1i H = H 22 h 2| | 2i

In simple gas-phase system (no environmental eﬀects), diagonal elements of Hamiltonian matrix described by: i i Hii = "ii = Uqq(R, Q)+↵gas

Potential Energy Interaction Gas Phase Shift Given by FF energy difference between inﬁnitely separated fragments.

9 Empirical Valence Bond (EVB)

i i Hii = "ii = Uqq(R, Q)+↵gas

Bond Stretch Bending and Torsional potentials

1 2 2 VkRR= − VD⎡⎤1 exp RR stretchR ( 0 ) stretch= e ⎣⎦−−(β ( 0 )) 1 2 2 Vk= θθ− VVsnstorsion =+n ⎡⎤1 cos( φ) ; =± 1 bend2 θ ( 0 ) ⎣⎦

1 2 Non-bonding potentials Vk= θθ− VVsnstorsion =+n ⎡⎤1 cos( φ) ; =± 1 bend2 θ ( 0 ) ⎣⎦ 2 2 ACAC 12 12 qqeijqqeij ijij ij ij AAAAijAAAA==ij== i j i; j; i i 4εσ i 4εσ i i i VCoulombVCoulomb= = VLennard-JonesVLennard-Jones= = 1212− − 6 6 RRRR 6 6 RijRij ijij ij ij CCCCijCCCC==ij== i j i; j; i i 4εσ i 4εσ i i i

Vsoft = Ci · C j · e(-ai ·a j ·ri, j )

10 Empirical Valence Bond (EVB)

i i Hii = "ii = Uqq(R, Q)+↵gas

1 " = M(b )+ K(1) (✓(1) ✓(1) )2 + U (1) + U (1) 11 1 2 ✓,m 0,m nb incact X 1 " = M(b )+ K(2) (✓(2) ✓(2) )2 + U (2) + U (2) + ↵i 22 2 2 ✓,m 0,m nb incact gas X H X C Y

H H

11 Empirical Valence Bond (EVB)

Off-diagonal Elements (the trickiest aspect….) H = H 12 h 1| | 2i

Barrier Height: one/two-dimensional Gaussian functions of interatomic distance.

0 2 Warshel, 1980 V12 = Aexp{−a(r1 − r1 ) }

0 2 0 0 0 2 V12 = Aexp{−(a(r1 − r1 ) + 2b(r1 − r1 )(r2 − r2 )+ c(r2 − r1 ) )}

Glowacki, 2011

Position and frequencies of the TS:

T T V12 = Aexp{B Δq − 0.5Δq ⋅C ⋅ Δq}, Δq = q − qTS

Chang-Miller, 1990

12 Empirical Valence Bond (EVB)

The ground-state potential surface of the system is obtained as the lowest eigenvalue of the 2 x 2 secular equation H " H "S 0= 11 12 12 H "S H " 21 21 22 Off-diagonals Elements Diagonal Elements H = H H11 = 1 H 1 12 h 1| | 2i h | | i H = H 22 h 2| | 2i

1 E = "" = (H + H ) (H + H )2 +4H 2 g 2 11 22 11 22 12 h p i 6 906 J. Chem. Theory Comput., Vol. 2, No. 4, 2006 Schlegel and Sonnenberg

NDim A ) [V11(qTS) - V(qTS)][V22(qTS) - V(qTS)] (6a) Chang & Miller, J. Phys. Chem. 1990, 94, 5884 V 2(q) ) B g(q,q ,i,j,R) (10) Jensen, J. Comput. Chem. 1994, 15, 1199. 12 ∑ ∑ ijK K K igjg0 D1 D2 Jensen, J. Am. Chem. Soc. 1992, 114, 1596. B ) + and Anglada, et al. J. Comput. Chem. 1999, 20,where 1112-1129. NDim is 3 times the number of atoms for a Cartesian [V11(qTS) - V(qTS)] [14V22(qTS) - V(qTS)] coordinate system or the number of coordinates if internal ∂V (q) ∂V(q) nn or redundant-internal coordinates are utilized. The Gaussian Dn ) q)q - q)q (6b) ∂q | TS ∂q | TS exponents are chosen such that the fit is sufficiently smooth for energies along the reaction path and V 2 is acceptably D D T D D T 12 ) 1 1 + 2 2 - small at the reactants and products, if these are not already C 2 2 [V11(qTS) - V(qTS)] [V22(qTS) - V(qTS)] included in qK. In the simplest approach, the exponents are

K1 K2 all equal; alternatively, if suitable criteria exist, they may - and be different for different centers, or even for different V11(qTS) - V(qTS) V22(qTS) - V(qTS) directions. The BijK coefficients are obtained by fitting to 2 2 ∂ Vnn(q) ∂ V(q) V 2 and its first and second derivatives at a number of points, K ) - (6c) 12 n 2 2 q , which can conveniently be the same as q . ∂q q)qTS ∂q q)qTS L K | | NDim The original version of the Chang-Miller method runs V 2(q ) ) B g(q ,q ,i,j,R) into difficulties when C has one or more negative eigenval- 12 L ∑ ∑ ijK L K K igjg0 12,15,18 2 ues. In these cases, the form of V12 in eq 5 diverges for large ∆q values. The simplest solution to this problem 2 NDim ∂V12 (q) ∂g(q,qK,i,j,R) switches the interaction matrix element to zero in regions ) ∑ ∑ BijK 2 15 q)q g g q)q where the unmodified V12 is negative or divergent. Another ∂q L K i j 0 ∂q L approach is to include suitable cubic and quartic terms in | | 2 2 NDim 2 the Gaussian to control asymptotic behavior.12 ∂ V12 (q) ∂ g(q,qK,i,j,R) 2 ) BijK (11) In the present article, an alternative form for V12 is pro- 2 ∑ ∑ 2 ∂q q)q K igjg0 ∂q q)q posed. Instead of using a generalized Gaussian as in eq 5, a | L | L quadratic polynomial times a spherical Gaussian is employed. If the number of Gaussian centers is equal to the number of 2 T 1 T points (i.e., if the number of coefficients is equal to the V (q) ) A[1 + B ‚∆q + / ∆q ‚(C +RI)‚∆q] 12 2 number of energy values, first derivatives, and second exp[-1/ R ∆q 2] (7) 2 | | derivatives), this is simply the solution of a set of linear equations. Fitting to the energy, gradient, and Hessian at the transition state yields the same formulas for A and B as those in the DB ) F (12) 906 J. Chem. Theory Comput., Vol. 2, No. 4, 2006 Schlegel and Sonnenberg 906 J. Chem. Theory Comput., Vol. 2, No. 4, 2006 Schlegel and Sonnenberg V11 ) 〈ψ1 Hˆ ψ1〉, V12 ) V21 ) 〈ψ1 Hˆ ψ2〉, Chang-Miller case; the expression for C is slightly different. V11 |) 〈|ψ1 Hˆ ψ1〉, V12 ) V21 ) 〈ψ1 Hˆ |ψ2〉|, Chang-Miller case; the expression for C is slightly different. | | | | V ) 〈ψ Hˆ ψ 〉 (3) T T V22 )22〈ψ2 Hˆ ψ2〉2 (3) 2 T T | | | | D1D2 + D2D1D1D2 +KD1 2D1 K1 C ) C ) + ++ + 1 1 2 2 A V (q ) - V(q ) 1V ) / (V + V ) - [ /1(V - V )] + V 2 (4) 2 11 ATS TS V (q ) - V(q ) V ) / (V 2 +11 V 22) - [ 2/ (11V -22 V )]12 + V (4) 11 TS TS 2 11 22 2 11 22 12 K2 ! (8) K Each matrix element is a function of molecular geometry, V (q ) - V(q ) 2 ! 22 TS TS (8) Each matrixq. Good element approximations is a for functionV11 and V of22 are molecular available from geometry, V (q ) - V(q ) molecular mechanics. However, much less is known about The exponent R is chosen to be small enough so that the 22 TS TS q. Good approximations for V11 and V22 are available from the functional form of the interaction matrix element, V12. PES is smooth but not so small that the reactant and product molecularIn mechanics. Warshel and Weiss’s However, approach, much2 the interaction less is known matrix aboutenergies are affectedThe exponent significantly.R Oneis approach chosen is to choose be small enough so that the the functionalelement V form12 is chosen of the to interaction reproduce the matrix barrier element, height VR 12to. give a goodPES fit is for smooth the energies but along not the so reaction small path. that the reactant and product 2 (obtained from experiments or calculations).2 For cases where The form of V12 in eq 7 can also be viewed as expanding In Warshel and Weiss’s approach, the interaction matrix2 energies are affected significantly. One approach is to choose greater accuracy is required, it is also desirable to match the V12 as a linear combination of s-, p-, and d-type Gaussians. elementpositionV12 andis vibrational chosen frequencies to reproduce of the TS the in addition barrier to height R to give a good fit for the energies along the reaction path. 1 2 3 the barrier height. The Chang-Miller approach describes g(q,qK,0,0,R) ) exp[- 2/2R q - qK ] (obtained from experiments or calculations). For cases where The form of V12 in| eq 7| can also be viewed as expanding the interactionDiﬀerent matrix element Flavours by a generalized of VB Gaussian 2 1 2 greater accuracy is required, it is also desirable to match theg(q,q ,i,0,VR12) )as(q a- linearq ) exp[ combination- / R q - q ] of s-, p-, and d-type Gaussians. positioned at or near the transition state K K i 2 | K| position and vibrational frequencies of the TS in addition to 1 2 2 T 1 T g(q,q ,i,j,R) ) (q - q ) (q - q ) exp[- / R q - q ] (9)1 2 V (q) ) A exp[B ‚∆q - / ∆q ‚C‚∆q], ∆q ) q - q3 K K i K j 2 K the barrier12 height. The Chang2 -Miller approachTS describes g(q,qK,0,0,R|) ) exp[| - /2R q - qK ] | | Chang-Miller theFormalism interaction matrix element by a generalized(5) GaussianBecause the coefficients in eq 7 are linear, the procedure where q is the transition-state geometry. The coefficients can be readily generalized to include Gaussians at multiple 1 2 TS g(q,qK,i,0,R) ) (q - qK)i exp[- /2R q - qK ] positionedare chosen at or so near that the the energy, transition gradient, state and second deriva- centers, qK. For example, one could choose to place the | | tives of the EVB surface match ab initio calculations at the Gaussian centers at the TS, reactant minimum, product 1 2 2 T 1 T g(q,q ,i,j,R) ) (q - q ) (q - q ) exp[- / R q - q ] (9) V (q)TS.) FollowingA exp[B Chang‚∆q--Miller’s/ ∆q notation,‚C‚∆3 thisq], yields∆q simple,) q - q minimum, and a fewK points along the reactionK i path to eitherK j 2 | K| 12 2 TS 2 closed-form equations for parameters A, B (a vector), and side of the transition state. The generalized form of V12 can C (a matrix). be(5) written asBecause the coefficients in eq 7 are linear, the procedure where qTS isA the) [ transition-stateV (q ) - V(q )][V geometry.(q ) - V(q The)] coefficients (6a) can be readilyNDim generalized to include Gaussians at multiple 11 TS TS 22 TS TS 2 are chosen so that the energy, gradient, and second deriva- centers,V12 (q) ) ∑qK∑. ForBijKg(q example,,qK,i,j,R) one (10) could choose to place the D D K igjg0 tives of the EVB surface1 match ab2 initio calculations at the Gaussian centers at the TS, reactant minimum, product B ) + and where NDim is 3 times the number of atoms for a Cartesian [V11(qTS) - V(qTS)] [V22(qTS) - V(qTS3)] TS. Following Chang-Miller’s notation, this yields simple,coordinate systemminimum, or the number and a of few coordinates points if along internal the reaction path to either ∂V (q) ∂V(q) nn or redundant-internal coordinates are utilized. The Gaussian 2 closed-form equations forDn ) parametersq)q -A, B q(a)q vector),(6b) and side of the transition state. The generalized form of V12 can ∂q | TS ∂q | TS exponents are chosen such that the fit is sufficiently smooth C (a matrix). be written as 2 T T for energies along the reaction path and V12 is acceptably D1D1 D2D2 C ) + - small at the reactants and products, if these are not already A ) [V (q ) - V2 (q )][V (q ) -2 V(q )] (6a) NDim [V11(q11TS) -TSV(qTS)] [TSV22(qTS)22- VTS(qTS)] TS included in qK. In the simplest approach,2 the exponents are all equal; alternatively, if suitableV criteria(q) ) exist, theyB mayg(q,q ,i,j,R) (10) K1 K2 12 ∑ ∑ ijK K D - D and be different for different centers, or evenK forigj differentg0 V111(qTS) - V(qTS) V22(qTS) - V2(qTS) B ) + and directions. The BijK coefficients are obtained by fitting to 2 2 where NDim is 3 times the number of atoms for a Cartesian [V (q ) - V(q )] ∂ [VVnn(q()q ) -∂VV(qq) )] 2 11 TS TS ) 22 TS - TS V12 and its first and second derivatives at a number of points, Kn 2 2 (6c) ∂q q)q ∂q q)q qL, which cancoordinate conveniently system be the same or as theqK. number of coordinates if internal ∂Vnn(q) TS ∂V(TSq) D ) | - | (6b) or redundant-internalNDim coordinates are utilized. The Gaussian The original versionn of the Chang-qMiller)qTS method runsq)qTS ∂q | ∂q | V 2(q ) ) B g(q ,q ,i,j,R) into difficulties when C has one or more negative eigenval- exponents12 L ∑ ∑ areijK chosenL K such that the fit is sufficiently smooth K igjg0 12,15,18 15 2 Chang & Miller, J. Phys. Chem.for 1990 energies, 94, 5884 along the reaction path and V 2 is acceptably ues. D DIn theseT cases, the form of DV12 DinT eq 5 diverges 12 for large 1∆q1values. The simplest solution2 to2 this problem ∂V 2(q) NDim ∂g(q,q ,i,j,R) C ) + - 12 small at the reactantsK and products, if these are not already switches the interaction2 matrix element to zero in regions2 ) ∑ ∑ BijK [V (q ) - V(q )] 2 [V (q ) - V(q15 )] includedq)q ing gqK. In the simplestq)q approach, the exponents are where11 TS the unmodifiedTS V12 is negative22 TS or divergent. TSAnother ∂q L K i j 0 ∂q L approach is to include suitable cubic and quartic terms in | | K1 K2 2 2 all equal; alternatively,2 if suitable criteria exist, they may 12 ∂ V (q) NDim ∂ g(q,q ,i,j,R) the Gaussian to control asymptotic- behavior. and 12 be different for differentK centers, or even for different V (q ) - V(q ) V (q ) - V(q 2 ) ) BijK (11) In11 the presentTS article,TS an alternative22 TS form for V12TSis pro- 2 ∑ ∑ 2 K igjg0 ∂q directions.q)qL The BijK∂qcoefficientsq)qL are obtained by fitting to posed. Instead of using a generalized2 Gaussian as2 in eq 5, a quadratic polynomial times∂ aV sphericalnn(q) Gaussian is∂ employed.V(q) V | 2 and its first and second| derivatives at a number of points, ) - If the number of12 Gaussian centers is equal to the number of Kn 2 2 (6c) 2 T 1 T points (i.e., ifqL the, which number can of coefficients conveniently is equal be to the the same as qK. V (q) ) A[1 + B ‚∆q + /∂∆qq ‚(C +Rq)qITS)‚∆q] ∂q q)qTS 12 2 number of energy values, first derivatives, and second 1 2 | exp[- / R ∆q ]| (7) NDim The original version of the Chang-Miller2 | | method runsderivatives), this is simply the solution of a set of linear equations. V 2(q ) ) B g(q ,q ,i,j,R) into difficultiesFitting to the when energy,C gradient,has one and or Hessian more at negative the transition eigenval- 12 L ∑ ∑ ijK L K K igjg0 12,15,18state yields the same formulas for A and B as2 those in the DB ) F (12) ues. In these cases, the form of V12 in eq 5 diverges for large ∆q values. The simplest solution to this problem 2 NDim ∂V12 (q) ∂g(q,qK,i,j,R) switches the interaction matrix element to zero in regions ) ∑ ∑ BijK 2 15 q)q g g q)q where the unmodified V12 is negative or divergent. Another ∂q L K i j 0 ∂q L approach is to include suitable cubic and quartic terms in | | 2 2 NDim 2 the Gaussian to control asymptotic behavior.12 ∂ V12 (q) ∂ g(q,qK,i,j,R) 2 ) BijK (11) In the present article, an alternative form for V12 is pro- 2 ∑ ∑ 2 ∂q q)q K igjg0 ∂q q)q posed. Instead of using a generalized Gaussian as in eq 5, a | L | L quadratic polynomial times a spherical Gaussian is employed. If the number of Gaussian centers is equal to the number of 2 T 1 T points (i.e., if the number of coefficients is equal to the V (q) ) A[1 + B ‚∆q + / ∆q ‚(C +RI)‚∆q] 12 2 number of energy values, first derivatives, and second exp[-1/ R ∆q 2] (7) 2 | | derivatives), this is simply the solution of a set of linear equations. Fitting to the energy, gradient, and Hessian at the transition state yields the same formulas for A and B as those in the DB ) F (12) 906 J. Chem. Theory Comput., Vol. 2, No. 4, 2006 Schlegel and Sonnenberg

V ) 〈ψ Hˆ ψ 〉, V ) V ) 〈ψ Hˆ ψ 〉, Chang-Miller case; the expression for C is slightly different. 11 1| | 1 12 21 1| | 2 V22 ) 〈ψ2 Hˆ ψ2〉 (3) T T | | D1D2 + D2D1 K1 C ) + + 1 1 2 2 A V11(qTS) - V(qTS) V ) /2(V11 + V22) - [ /2(V11 - V22)] + V12 (4) K ! 2 Each matrix element is a function of molecular geometry, (8) V22(qTS) - V(qTS) q. Good approximations for V11 and V22 are available from molecular mechanics. However, much less is known about The exponent R is chosen to be small enough so that the the functional form of the interaction matrix element, V12. PES is smooth but not so small that the reactant and product In Warshel and Weiss’s approach,2 the interaction matrix energies are affected significantly. One approach is to choose element V12 is chosen to reproduce the barrier height R to give a good fit for the energies along the reaction path. 2 (obtained from experiments or calculations). For cases where The form of V12 in eq 7 can also be viewed as expanding 2 greater accuracy is required, it is also desirable to match the V12 as a linear combination of s-, p-, and d-type Gaussians. position and vibrational frequencies of the TS in addition to 1 2 3 the barrier height. The Chang-Miller approach describes g(q,qK,0,0,R) ) exp[- /2R q - qK ] | | the interaction matrix element by a generalized Gaussian g(q,q ,i,0,R) ) (q - q ) exp[-1/ R q - q 2] positioned at or near the transition state K K i 2 | K| 1 2 2 T 1 T g(q,qK,i,j,R) ) (q - qK)i(q - qK)j exp[- /2R q - qK ] (9) V12 (q) ) A exp[B ‚∆q - /2∆q ‚C‚∆q], ∆q ) q - qTS | | (5) Because the coefficients in eq 7 are linear, the procedure where qTS is the transition-state geometry. The coefficients can be readily generalized to include Gaussians at multiple are chosen so that the energy, gradient, and second deriva- centers, qK. For example, one could choose to place the tives of the EVB surface match ab initio calculations at the Gaussian centers at the TS, reactant minimum, product TS. Following Chang-Miller’s notation,3 this yields simple, minimum, and a few points along the reaction path to either 2 closed-form equations for parameters A, B (a vector), and side of the transition state. The generalized form of V12 can C (a matrix). be written as A ) [V (q ) - V(q )][V (q ) - V(q )] (6a) NDim 11 TS TS 22 TS TS 2 V12 (q) ) ∑ ∑ BijKg(q,qK,i,j,R) (10) K igjg0 D1 D2 B ) + and where NDim is 3 times the number of atoms for a Cartesian [V11(qTS) - V(qTS)] [V22(qTS) - V(qTS)] 906 J. Chem. Theory Comput., Vol. 2, No. 4, 2006 906 J. Chem. Theory Comput.,Schlegel Vol. and Sonnenberg 2, No. 4, 2006 coordinate system or the numberSchlegel of coordinates and Sonnenberg if internal ∂V (q) ∂V(q) nn or redundant-internal coordinates are utilized. The Gaussian Dn ) q)q - q)q (6b) V11 ) 〈ψ1 Hˆ ψ1〉, V12 ) V21 ) 〈ψ1 Hˆ ψ2〉, Chang-MillerV11 case;) 〈ψ the1 Hˆ expressionψ1〉, V12 ) forVC21is) slightly∂〈qψ1 Hˆ different.ψTS2〉, ∂q TS Chang-Miller case; the expression for C is slightly different. | | | | | | || | | exponents are chosen such that the fit is sufficiently smooth V ) 〈ψ Hˆ ψ 〉 (3) T T V ) 〈ψ Hˆ ψ 〉 (3) 2 22 2| | 2 D D + D D TK 22T 2| | 2 forD energiesD T + D alongD T the reactionK path and V12 is acceptably 1 2 2 1 D1D1 1 D2D2 1 2 2 1 1 C ) C ) + + + - Csmall) at the reactants+ and products, if these+ are not already 1 1 2 2 A 1V11(qTS) - V(q2 TS) 1 2 2 2 A V11(qTS) - V(qTS) V ) /2(V11 + V22) - [ /2(V11 - V22)] + V12 (4) [VV)(q/ (V) - +V(Vq ))]- [V/ ((Vq -) -VV)](q +)]V (4) included in qK. In the simplest approach, the exponents are 11 TS2 11 TS22 222 K 11TS 22 TS 12 2 all equal; alternatively, if suitable criteriaK exist,2 they may ! K1 ! K2 (8) (8) Each matrix element is a function of molecular geometry, Each matrix element is aV function-(q ) - ofV( molecularq ) and geometry, V (q ) - V(q ) V (q ) - V(q )22 VTS (q ) -TSV(q ) be different for different centers, or22 evenTS for differentTS q. Good approximations for V11 and V22 are available from q. Good approximations11 TS TS for V11 22andTSV22 are availableTS from directions. The BijK coefficients are obtained by fitting to molecular mechanics. However, much less is known about The exponentmolecularR is chosen mechanics. to be small However,2 enough much so less that is the2 known about ∂ Vnn(q) ∂ V(q) TheV 2 exponentand its firstR andis chosen second to derivatives be small at enough a number so thatof points, the K ) - (6c) 12 the functional form of the interaction matrix element, V12. PES is smooththe but functional not so small form that ofn the the interactionreactant2 and matrix product element,2 V12. PES is smooth but not so small that the reactant and product ∂q q)q ∂q q)q qL, which can conveniently be the same as qK. In Warshel and Weiss’s approach,2 the interaction matrix energies are affectedIn Warshel significantly. and Weiss’s One approachapproach, is2 the toTS choose interaction matrixTS energies are affected significantly. One approach is to choose | | NDim element V12 is chosen to reproduce the barrier height R to give aelement good fit forV12 theis energies chosen along to reproduce the reaction the path. barrier height The original version of the Chang-Miller method runs R to give a good2 fit for the energies along the reaction path. (obtained from experiments or calculations). For cases where 2 V 2 (q ) ) B g(q ,q ,i,j,R) The form of(obtainedintoV12 difficultiesin eq from 7 can experiments when alsoC behas viewed or one calculations). or as more expanding negative For cases eigenval- where The form of V1212 inL eq 7∑ can∑ alsoijK be viewedL K as expanding greater accuracy is required, it is also desirable to match the V 2 as a linear combinationDiff oferent s-, p-, Flavours and d-type Gaussians. of VB 2 K igjg0 12 greaterues.12,15,18 accuracyIn these is required, cases, the it is form also of desirableV 2 in toeq match 5 diverges the V12 as a linear combination of s-, p-, and d-type Gaussians. position and vibrational frequencies of the TS in addition to 12 position and vibrational1 frequencies2 of the TS in addition to 2 3 for large ∆q values. The simplest solution to this problem ∂V (q) NDim 1 ∂g(q,q ,i,j,2R) the barrier height. The Chang-Miller approach describes g(q,qK,0,0,R) ) exp[- /2R q - qK ] 3 g12(q,q ,0,0,R) ) exp[- / R q -Kq ] theswitches barrier the height. interaction The Chang| matrix-Miller element| approach to zerodescribes in regions K 2 K ) ∑ ∑ BijK | | the interaction matrix element by a generalized Gaussian Chang-Miller theFormalism interaction matrix element1 2 by a2 generalized Gaussian15 ∂ q)q g g ∂ q)q g(q,q ,i,0,whereR) ) the(q unmodified- q ) exp[-V12/ Risq negative- q ] or divergent. Another q L K i j 0 1 q 2 L positioned at or near the transition state K K i 2 K g(q,qK,i,0,R) ) (q - qK)i exp[- /2R q - qK ] positionedapproach at is or to near include the suitable transition| cubic state| and quartic terms in | | || 1 2 2 2 2 12 ∂ NDim ∂ 1 2 2 T 1 T g(q,qK,i,j,R) the)2(q Gaussian- qK)i(q to- controlqT K)j exp[ asymptotic1- /2RT q - behavior.qK ] (9) g(q,q ,i,j,RV12) )(q()q - q ) (q - q ) exp[g-(q,/qRK,iq,j,-R)q ] (9) V12 (q) ) A exp[B ‚∆q - /2∆q ‚C‚∆q], ∆q ) q - qTS V (q) ) A exp[B ‚∆q - / ∆q |‚C‚∆q],| ∆q ) q - q K K i K j 2 K 12 2 2 TS ) BijK | | (11) In the present article, an alternative form for V12 is pro- 2 ∑ ∑ 2 (5) Because the coefficients in eq 7 are linear, the procedure ∂q q)q K igjg0 ∂q q)q posed. Instead of using a generalized Gaussian as in eq(5) 5, a Because the coefficientsL in eq 7 are linear, the procedureL where qTS is the transition-state geometry. The coefficients can be readily generalized to include Gaussians at multiple | | wherequadraticqTS is polynomial the transition-state times a spherical geometry. Gaussian The coefficients is employed. can be readily generalized to include Gaussians at multiple are chosen so that the energy, gradient, and second deriva- Modifiedcenters, Chang-Millerq . For example, Formalism one could choose to place the If the number of Gaussian centers is equal to the number of Kare chosen so that the energy, gradient, and second deriva- centers, qK. For example, one could choose to place the 2 T 1 T points (i.e., if the number of coefficients is equal to the tives of the EVB surface match ab initio calculations at the Gaussian centerstivesV ( ofq at) the) theA EVB[1 TS,+ surfaceB reactant‚∆q + match minimum,/ ∆q ab‚( initioC +R product calculationsI)‚∆q] at the Gaussian centers at the TS, reactant minimum, product 3 12 2 - 3 number of energy values, first derivatives, and second TS. Following Chang Miller’s notation, this yields simple, minimum, andTS. a Following few points Chang along- theMiller’s reaction notation, path tothis either yields1 simple,2 minimum, and a few points along the reaction path to either exp[2 - / R ∆q ] (7) closed-form equations for parameters A, B (a vector), and side of the transition state. The generalized form of V12 can 2 derivatives), this is simply the solution of a set of2 linear closed-form equations for parameters A, B (a vector),| | and side of the transition state. The generalized form of V12 can C (a matrix). be written as equations. Multiple GuassiansCFitting(a matrix). to the energy, gradient, and Hessian at the transition be written as NDim A ) [V (q ) - V(q )][V (q ) - V(q )] (6a) state yields the same formulas for A and B as those in the NDimDB ) F (12) 11 TS TS 22 TS TS 2 A ) [V (q ) - V(q )][V (q ) - V(q )] (6a) 11 TS TS 22 TS TS 2 V12 (q) ) BijKg(q,qK,i,j,R) (10) ∑ ∑ NDim is 3 x Natoms V12 (q) ) BijKg(q,qK,i,j,R) (10) K igjg0 ∑ ∑ D1 D2 K igjg0 D1 D2 B ) + and where NDimB is) 3 times the number of+ atoms for a Cartesianand [V11(qTS) - V(qTS)] [V22(qTS) - V(qTS)] where NDim is 3 times the number of atoms for a Cartesian coordinate system[V11 or(q theTS) number- V(qTS)] of coordinates[V22(qTS) - ifV internal(qTS)] ∂Vnn(q) ∂V(q) coordinate system or the number of coordinates if internal or redundant-internal coordinates are utilized.∂Vnn(q The) Gaussian∂V(q) Dn ) q)q - q)q (6b) or redundant-internal coordinates are utilized. The Gaussian ∂q TS ∂q TS Dn ) q)q - q)q (6b) | | exponents are chosen such that the fit is sufficiently∂ TS smooth∂ TS q | Changq | & Miller, J. Phys. Chem. 1990, 94, 5884 2 exponents are chosen such that the fit is sufficiently smooth T T for energies along the reaction path and16 V12 is acceptablySchlegel J. Chem. Theory Comput. 2006, 2, 905 2 D1D1 D2D2 T T for energies along the reaction path and V12 is acceptably C ) + - small at the reactants andD1D products,1 if these areD not2D2 already 2 2 C ) + - small at the reactants and products, if these are not already [V11(qTS) - V(qTS)] [V22(qTS) - V(qTS)] included in qK. In the simplest approach,2 the exponents are 2 [V11(qTS) - V(qTS)] [V22(qTS) - V(qTS)] included in qK. In the simplest approach, the exponents are K1 K2 all equal; alternatively, if suitable criteria exist, they may - and K1 K2 all equal; alternatively, if suitable criteria exist, they may be different for different centers, or- even for different and V11(qTS) - V(qTS) V22(qTS) - V(qTS) be different for different centers, or even for different directions. The BijK Vcoefficients11(qTS) - V( areqTS) obtainedV22(qTS by) - fittingV(qTS to) 2 2 ∂ V (q) 2 2 directions. The BijK coefficients are obtained by fitting to nn ∂ V(q) V12 and its first and second derivatives at a number of points,2 ) - ∂ Vnn(q) ∂ V(q) 2 Kn 2 2 (6c) V12 and its first and second derivatives at a number of points, ∂ q)q ∂ q)q qL, which can conveniently beK then ) same as qK. - (6c) q TS q TS 2 2 q , which can conveniently be the same as q . ∂q q)qTS ∂q q)qTS L K | | NDim The original version of the Chang-Miller method runs | | NDim VThe2(q original) ) versionB g( ofq , theq ,i, Changj,R) -Miller method runs into difficulties when C has one or more negative eigenval- 12 L ∑ ∑ ijK L K V 2(q ) ) B g(q ,q ,i,j,R) into difficultiesK ig whenjg0 C has one or more negative eigenval- 12 L ∑ ∑ ijK L K 12,15,18 2 K igjg0 ues. In these cases, the form of V12 in eq 5 diverges 12,15,18 2 ues. In these cases, the form of V12 in eq 5 diverges for large ∆q values. The simplest solution to this problem ∂V 2(q) NDim ∂g(q,q ,i,j,R) 12for large ∆q values. The simplestK solution to this problem 2 NDim ∂V12 (q) ∂g(q,qK,i,j,R) switches the interaction matrix element to zero in regions ) BijK 2 15 switches) the∑ interaction∑ matrix element) to zero in regions ) B ∂q q qL K igjg0 ∂q q qL ∑ ∑ ijK where the unmodified V12 is negative or divergent. Another 2 15 q)q q)q where the unmodified V is negative or divergent. Another ∂q L K igjg0 ∂q L approach is to include suitable cubic and quartic terms in | 12 | 2 2 2 | | 12 ∂ V approach(q) is toNDim include∂ suitableg(q,q ,i, cubicj,R) and quartic terms in the Gaussian to control asymptotic behavior. 12 K 2 2 NDim 2 12 ∂ V12 (q) ∂ g(q,qK,i,j,R) 2 the Gaussian) to controlBijK asymptotic behavior.(11) In the present article, an alternative form for V12 is pro- 2 ∑ ∑ 2 K igjg0 2 ) BijK (11) ∂q Inq the)qL present article, an alternative∂q q) formqL for V12 is pro- 2 ∑ ∑ 2 posed. Instead of using a generalized Gaussian as in eq 5, a ∂q q)q K igjg0 ∂q q)q posed.| Instead of using a generalized| Gaussian as in eq 5, a L L quadratic polynomial times a spherical Gaussian is employed. If the number of Gaussian centers is equal to the number of | | quadratic polynomial times a spherical Gaussian is employed. If the number of Gaussian centers is equal to the number of 2 T 1 T points (i.e., if the number of coefficients is equal to the V12 (q) ) A[1 + B ‚∆q + /2∆q ‚(C +RI)‚∆q] 2 T 1 T points (i.e., if the number of coefficients is equal to the number ofV energy(q) ) values,A[1 + B first‚∆q derivatives,+ / ∆q ‚(C and+R secondI)‚∆q] 1 2 12 2 number of energy values, first derivatives, and second exp[- /2R ∆q ] (7) derivatives), this is simply the solution of a set of linear1 2 | | exp[- / R ∆q ] (7) equations. 2 | | derivatives), this is simply the solution of a set of linear Fitting to the energy, gradient, and Hessian at the transition equations. state yields the same formulas for A and B as those in the Fitting to the energy,DB ) F gradient, and Hessian(12) at the transition state yields the same formulas for A and B as those in the DB ) F (12) Different Flavours of VB

Empirical Valence-Bond Models J. Chem. Theory Comput., Vol. 2, No. 4, 2006 907 Intersecting Morse Curves Chang-Miller EVB

Modiﬁed Chang-Miller EVB

Figure 1. (a) One-dimensional potentialWarshel energy EVB curve (solid line) constructed fromMultiple two interacting Gaussian Morse curves (chain-dot). V11 and V22 (long dash) are quadratic functions fitted to the minima of the Morse curves, fitted using various EVB models (short 2 2 dash). (b) EVB model with constant V12 . (c) Chang-Miller EVB model with V12 represented by a generalized Gaussian. (d) EVB model with V 2 represented by a quadratic polynomial times a Gaussian. (e) EVBChang model & Miller, with a J. three-Gaussian Phys. Chem. fit. 1990, 94, 5884 12 17 Schlegel2 J. Chem. Theory Comput. 2006, 2, 905 where D is a matrix containing the values of g(qL,qK,i,j,R), matrix element, V12 ) 0.010 au. The empirical valence bond 2 2 ∂g(qL,qK,i,j,R)/∂q q)qL, and ∂ g(qL,qK,i,j,R)/∂q q)qL and F is approximation to the surface is constructed from two | 2 | 2 a column vector containing the values of V12 (qL), ∂V12 (q)/ quadratic potentials fitted to the individual Morse functions 2 2 2 ∂q ) ,and∂ V (q)/∂q ) .Evenwithonlyafewexpan- at their minima. As can be seen from Figure 1a, in the region |q qL 12 |q qL sion centers, the eigenvalues of D become very small because of the transition state, V11 and V22 are much higher than the 2 of strong overlap between the Gaussians. In this case, the potential energy curve being modeled. Hence, V12 will have coefficients can be chosen in a least-squares manner. Sim- to be quite large, providing a suitable challenge for the ilarly, if the number of Gaussian centers in the expansion is methodology. 2 2 2 chosen to be smaller than the number of points where V12 Starting with Warshel and Weiss’s method, V12 is set and its derivatives are evaluated, then the coefficients can equal to a constant. In Figure 1b, the constant is chosen to also be obtained in a least-squares manner. reproduce the forward barrier height, and the curve is dis-

T placed to match the energies of the reactant and TS. Note minimize(DB - F) W(DB - F) that the resulting minima positions are shifted, the barrier DTWDB ) DTWF (13) width is too small, and the reaction exothermicity is too large. With only one parameter, fitting the potential energy curve where W is a diagonal weighting matrix. This can be solved well is difficult. easily using singular value decomposition. 2 In the Chang-Miller approach, V12 is represented by a Examples generalized Gaussian, with the parameters fitted to the tran- One-Dimensional Test CasesIntersecting Morse CurVes. A sition-state energy, gradient, and Hessian. For this example, simple one-dimensional potential energy curve can be con- the parameters for eq 5 are A ) 0.167, B ) 0.385, and C ) structed from two intersecting Morse curves, as shown in 1.988. As shown in Figure 1c, this yields a significant im- 2 Figure 1a. This resembles the potential energy along the provement in fit to the potential energy curve. Because V12 reaction path for hydrogen abstraction reactions, X-H + Y is not zero at the reactant and product geometries, the minima f X + H-Y, and similar atom-transfer processes involving are slightly displaced, though not as much as in Figure 1b. 2 the forming and breaking of single bonds. The parameters When V12 is represented by a single Gaussian times a qua- for the Morse curves are De ) 0.12 and 0.16 au with force dratic polynomial, Figure 1d, the results are similar to the constants at the minima of 0.40 and 0.50 au, respectively; Chang-Miller approach. The Gaussian exponent R can be the curves are displaced by 3.00 au and interact by a small varied over the range 1.5-3.0 (bracketing the Chang-Miller Empirical Valence Bond (EVB)

Summary

H " H "S 0= 11 12 12 H "S H " 21 21 22 1) Calculate H11/H12 2) Fit H12 to reproduce experimental or (QM) data

3) Eigenvalues are obtained by diagonalising the energy 2 x 2 matrix. The minimal value is the ground state.

1 E = "" = (H + H ) (H + H )2 +4H 2 g 2 11 22 11 22 12 h p i

6 Empirical Valence Bond (EVB)

Solvent eﬀects

ˆ ii i i i Hii = εi =Uqq (R,Q)+U pq (R,Q,r, q)+U pp (r, q)+αgas

Now also include interaction between reacting fragments and atoms of environment (p with charges q)

Also include interaction of all environment atoms within themselves. gas-phase shift is phase independent, and oﬀ-diagonal elements remain same for any electrostatic environment.

19 Empirical Valence Bond (EVB)

Solvent eﬀects H12 = (ε1 − Eg )(ε2 − Eg )

- - Cl + CH3 – Cl Cl – CH3 + Cl

20 Hong, et al J. Phys. Chem. B (2006), 110, 19570 From Potential Energy to Free Energy

So far we have only considered how to obtain the potential energy surface for a given reaction….

.. However, we want to obtain the Gibbs free energy (NpT)

Free-energy perturbation (FEP) - Umbrella sampling (US)

δG(1→ 2) = −kBT ln exp −(ε2 −ε1) / kBT { } 1

Although the expression is exact, it converges very slowly if the states are far away from each other in conﬁgurational space.

21 From Potential Energy to Free Energy

Mapping potential εm = (1− λm )ε1 + λmε2 (λm = 0, 1/N, 2/N, …, N/N)

This mapping potential can drive the system through a “non-physical” process

The free energy associated to this process is:

22 From Potential Energy to Free Energy

X- C Y X C Y X C Y-

24.0

18.0 ΔG(λ)

(kcal/mol)

) 12.0

λ G(

Δ 6.0

0.2 0.4 0.6 0.8 λ is not sufﬁcient for evaluating the activation free energy

23 From Potential Energy to Free Energy

" =(1 )" + " Mapping potential m m 1 m 2

1 Potential Energy E = " = [(H + h ) (H + H )2 +4H2 ] g 2 11 22 11 22 12 q

n n Reaction Coordinate (X) in terms of energy gap "(s )=X

sn: subspaces of the conﬁguration space (bins)

exp G(Xn) = exp G( ) exp (E (Xn) " (Xn)) { } { m }h { g m }im

24 Dynamics of SN2 Reactions in Solution J. Am. Chem. Soc., Vol. 110, No. 16, 1988 5303

i\ - I A-u-& 24.00 h4 s 80.00 '18.00 xc / .o vz 60.00 - 12.00 L- From Potential Energy to Free Energy Y d 9 y 40.00 - v ba 4 n n n exp G(X )20.00 = exp G( ) exp (E (X ) " (X )) 1 m Ip g m m 0.20 0.40 0.60 0.80 { } { }h { }i h -150.00 -50 00 50.00 150.00 Figure 6. Parametric dependence of the mapping free energy on the A&12(kcal/mol) parameter A. The figure shows both the total free energy AG(X) (solid line) and the electrostatic contribution AG,,(X) (dotted line). The figure FigureX- 7. FreeC energyY functionals,X Ag, andC Ag2 (seeY also ref 16c),X as a C Y- demonstrates that the largest contribution to AG(X) is due to electrostatic function of the energy gap Ae. The calculations are done with (0)and interactions. without (A)solvent-induced dipoles. configuration space. The EVB method provides a very convenient I ..&%.. mapping potential for doing this, and the free energy perturbation method tells us how to evaluate the free energy change associated with the mapping process (see below). Unfortunately, none of our mapping potentials coincide with the actual ground-state potential. Thus, we take a two-step strategy starting with a convenient mapping procedure and then using the relevant raw data to construct the ground-state free energy surface. We start by performing a series of simulations using a potential of the form (18) and changing the vector A" = (XIm,A*'") in a systematic way from (1, 0) to (0, 1) in n small increments, where the index m is changed -8 2'oo from 0 to n - 1. In this way we can drive the system from the reactant 1I - e, = XIm(, + h2'"e2 - 2(XlmX2m)1/2(H121 Xlm + X2"' 1 (18) -144 00 -48 00 48.00 144.00 A&u(kcal/mol) state through the transition state and into the product state. Note that Figure 8. Actual free energy profile for25 the ground-stateWarshel surface & Weiss, as J. a Am. Chem. Soc. 1980, 102, 6218 el has a minimum at the reactant geometry and e2 has a minimum at the function of the energy gap Ae. The calculations are done for the CI- + product geometry so that as we change m,we force the system to move CHQ-* CICH, + CI- exchange reaction, with (0)and without (A) from the reactant state to the product state. Specifically, the solvent is solvent-induced dipoles. forced to adjust its polarization to the changing charge distribution of the solute, and we will be able to see how much free energy the solvent responding formulation (which is described in detail elsewherei6CJ8c) needs to adjust to the solute configuration by using the free energy defines the reaction coordinate Xn in terms of the energy gap t2 - el. This perturbation formalism described below. is done by dividing the configuration space of the system into subspaces, The HI, term, which was not used in our previous studies, minimizes S", that satisfy the relationship At@") = X", in which the energy dif- the difference between the mapping potential and the actual ground-state ferences Xnare constants and Ae = e2 - el. The X' are then used as the potential. That is, the convergence of the calculated ground-state free reaction coordinate, giving energy depends (see below) on a correction term involving the difference between E, and e,. Thus, we use an em that is as close as possible to E,. exp[-Ag(X")PI = ex~[-AG(h" (exp[-(Eg(X") - cm(X"))Pl ) (2 1) Unfortunately, the ground-state potential is given by the complicated function (eq 8 and 19), where the C,'s are implicit functions of the Hiis. This expression relates the probability of finding the system at X" on the ground state E, in terms of the probability of being on the mapping potential e, that keeps X around X". The evaluation of Ag(X") for an exchange reaction is described in Figure 8. In addition to Ag(Xn), we As the simulation proceeds, the eigenvector C, fluctuates in an unpre- can obtain the probability of being at X" on el. This probability defines dictable way, making it hard to use it for generating a mapping potential. the free energy functionals, Agi (Agi(Xn) = .fexp[-ei(Xn)p] dS"), which However, replacing C, by a constant vector C,, with C,, = (Xim)1/2, can be obtained through the same derivation that led to eq 21, and are produces (with XI = X2 = 0.5) a potential e, that approaches E, at the given by (22). The evaluation of the Ag, functionals for an exchange transition-state region, where C,, N Cs2= 1/(2Il2). This guarantees reaction is demonstrated in Figure 7. stable results at the transition-state region. The free energy associated with changing el to e2 may be obtained with the relation~hip~~.~~(20), where ( )m indicates an average of the quantity This work attempts to evaluate free energy relationships on a micro- within the brackets calculated on potential surface e,, and the free energy scopic level (the actual calculations are described in section 111). To aG(Xm -* A,) = -(I /P) In [(exp(-(c,, - €m))P)ml (20) examine simple analytical approximations for the calculated dependence of Ag* on AGO, we consider the harmonic expansion of our diabatic *I surface. This expansion can be obtained from the power spectra of the AG(X,) = AG(Xo - A,) = C 6G(Xm - Am+,) mi0 time-dependent c,(r) as was done in our dispersed polaron and gives (23), where the Rsand Qs are the dimensionless coordinates for functional AG(X) reflects both the electrostatic solvation effects associated with the changes of the solute charges as well as the intramolecular el E E(h/Z)w,'(r, + A:/2)2 + E(h/2)wd(q, + A;/2)2 E effects associated with changing el to el. Figure 6 demonstrates the I J dependence of AG(h) on X for an exchange reaction where X = Y. The (h/Z)w,(R + A,/2l2 + (h/2)w~(Q+ A./2)2 (23) figure gives both the total AG(X) and its electrostatic components. t2 = x(h/2)w,'(rl - A,'/2) + C(h/Z)w;(% - A4/2)2 + AVO N Obtaining AG(X) with the perturbation procedure using the mapping I potentials em is not sufficient for evaluating the activation free energy, (h/z)w~(R- AR/~)* + (h/2)w~(Q- AQ/~)+ AVO Ag*, which reflects the probability of being at the transition state on the actual ground-state potential surface. The quantity AG(X) is basically the solute and the solvent, respectively. The effective frequencies up and used to find the free energy associated with changing el to the potential wR are evaluated by w = .f;w P(w) dw in which P(w) is the normalized em* that forces the system to spend the longest time near the actual power spectrum of the corresponding contribution to (e2 - el). The R are transition state. However, one still needs to find the probability of related to the regular reaction coordinate R' = (6, - b2) by R = R'- reaching the transition state for trajectories that move on E,. The cor- (wR/~R/h)l/'where pR is the reduced mass for the normal mode that is From Potential Energy to Free Energy

Dynamics of SN2 Reactions in Solution exp G (Xn) J.= Am.exp Chem.G( Soc.,) Vol.exp 110,(✏ No.(X n16,) 1988" (X n5303)) { i } { m }h { i m }im i\ - I A-u-& 24.00 h4 s 80.00 '18.00 xc / .o vz 60.00 - 12.00 L- Y d 9 y 40.00 - v ba 4

20.00 1 Ip 0.20 0.40 0.60 0.80 h -150.00 -50 00 50.00 150.00 Figure 6. Parametric dependence of the mapping free energy on the A&12(kcal/mol) parameter A. The figure shows both the total free energy AG(X) (solid line) and the electrostatic contribution AG,,(X) (dotted line). The figure Figure 7. Free energy functionals, Ag, and Ag2 (see also ref 16c), as a demonstrates that the largest contribution to AG(X) is due to electrostatic function of the energy gap Ae. The calculations are done with (0)and 26 Warshel & Weiss, J. Am. Chem. Soc. 1980, 102, 6218 interactions. without (A)solvent-induced dipoles. configuration space. The EVB method provides a very convenient I ..&%.. mapping potential for doing this, and the free energy perturbation method tells us how to evaluate the free energy change associated with the mapping process (see below). Unfortunately, none of our mapping potentials coincide with the actual ground-state potential. Thus, we take a two-step strategy starting with a convenient mapping procedure and then using the relevant raw data to construct the ground-state free energy surface. We start by performing a series of simulations using a potential of the form (18) and changing the vector A" = (XIm,A*'") in a systematic way from (1, 0) to (0, 1) in n small increments, where the index m is changed -8 2'oo from 0 to n - 1. In this way we can drive the system from the reactant 1I - e, = XIm(, + h2'"e2 - 2(XlmX2m)1/2(H121 Xlm + X2"' 1 (18) -144 00 -48 00 48.00 144.00 A&u(kcal/mol) state through the transition state and into the product state. Note that Figure 8. Actual free energy profile for the ground-state surface as a el has a minimum at the reactant geometry and e2 has a minimum at the function of the energy gap Ae. The calculations are done for the CI- + product geometry so that as we change m,we force the system to move CHQ-* CICH, + CI- exchange reaction, with (0)and without (A) from the reactant state to the product state. Specifically, the solvent is solvent-induced dipoles. forced to adjust its polarization to the changing charge distribution of the solute, and we will be able to see how much free energy the solvent responding formulation (which is described in detail elsewherei6CJ8c) needs to adjust to the solute configuration by using the free energy defines the reaction coordinate Xn in terms of the energy gap t2 - el. This perturbation formalism described below. is done by dividing the configuration space of the system into subspaces, The HI, term, which was not used in our previous studies, minimizes S", that satisfy the relationship At@") = X", in which the energy dif- the difference between the mapping potential and the actual ground-state ferences Xnare constants and Ae = e2 - el. The X' are then used as the potential. That is, the convergence of the calculated ground-state free reaction coordinate, giving energy depends (see below) on a correction term involving the difference between E, and e,. Thus, we use an em that is as close as possible to E,. exp[-Ag(X")PI = ex~[-AG(h" (exp[-(Eg(X") - cm(X"))Pl ) (2 1) Unfortunately, the ground-state potential is given by the complicated function (eq 8 and 19), where the C,'s are implicit functions of the Hiis. This expression relates the probability of finding the system at X" on the ground state E, in terms of the probability of being on the mapping potential e, that keeps X around X". The evaluation of Ag(X") for an exchange reaction is described in Figure 8. In addition to Ag(Xn), we As the simulation proceeds, the eigenvector C, fluctuates in an unpre- can obtain the probability of being at X" on el. This probability defines dictable way, making it hard to use it for generating a mapping potential. the free energy functionals, Agi (Agi(Xn) = .fexp[-ei(Xn)p] dS"), which However, replacing C, by a constant vector C,, with C,, = (Xim)1/2, can be obtained through the same derivation that led to eq 21, and are produces (with XI = X2 = 0.5) a potential e, that approaches E, at the given by (22). The evaluation of the Ag, functionals for an exchange transition-state region, where C,, N Cs2= 1/(2Il2). This guarantees reaction is demonstrated in Figure 7. stable results at the transition-state region. The free energy associated with changing el to e2 may be obtained with the relation~hip~~.~~(20), where ( )m indicates an average of the quantity This work attempts to evaluate free energy relationships on a micro- within the brackets calculated on potential surface e,, and the free energy scopic level (the actual calculations are described in section 111). To aG(Xm -* A,) = -(I /P) In [(exp(-(c,, - €m))P)ml (20) examine simple analytical approximations for the calculated dependence of Ag* on AGO, we consider the harmonic expansion of our diabatic *I surface. This expansion can be obtained from the power spectra of the AG(X,) = AG(Xo - A,) = C 6G(Xm - Am+,) mi0 time-dependent c,(r) as was done in our dispersed polaron and gives (23), where the Rsand Qs are the dimensionless coordinates for functional AG(X) reflects both the electrostatic solvation effects associated with the changes of the solute charges as well as the intramolecular el E E(h/Z)w,'(r, + A:/2)2 + E(h/2)wd(q, + A;/2)2 E effects associated with changing el to el. Figure 6 demonstrates the I J dependence of AG(h) on X for an exchange reaction where X = Y. The (h/Z)w,(R + A,/2l2 + (h/2)w~(Q+ A./2)2 (23) figure gives both the total AG(X) and its electrostatic components. t2 = x(h/2)w,'(rl - A,'/2) + C(h/Z)w;(% - A4/2)2 + AVO N Obtaining AG(X) with the perturbation procedure using the mapping I potentials em is not sufficient for evaluating the activation free energy, (h/z)w~(R- AR/~)* + (h/2)w~(Q- AQ/~)+ AVO Ag*, which reflects the probability of being at the transition state on the actual ground-state potential surface. The quantity AG(X) is basically the solute and the solvent, respectively. The effective frequencies up and used to find the free energy associated with changing el to the potential wR are evaluated by w = .f;w P(w) dw in which P(w) is the normalized em* that forces the system to spend the longest time near the actual power spectrum of the corresponding contribution to (e2 - el). The R are transition state. However, one still needs to find the probability of related to the regular reaction coordinate R' = (6, - b2) by R = R'- reaching the transition state for trajectories that move on E,. The cor- (wR/~R/h)l/'where pR is the reduced mass for the normal mode that is Diﬀerent Flavours of VB

Extended Method VB EVB Miller-EVB AVB MS-EVB MC-MM EVB Chang Vuilleumier Author Coulson Warshel McCammon Voth Truhlar & Miller & Borgis

Year 1954 1980 1991 1996 1997 1998 2001

H12 Exponential Exponential Exponential Constant Exponential General General

function Morse Morse+MM Morse+MM Morse+MM Morse+MM Morse+MM MM Solvent No In Hii No In Hii In Hii In Hii and Hij In Hii

27 Examples: Enzyme Catalysis

Phosphonate Monoester Phosphate Diester O O O O P P O O NO2 O NO2

Phosphate Monoester Sulfate Monoester O O O O P S O O O NO2 O NO2

Sulfonate Monoester O O S I

O NO2

What is the molecular origin of promiscuity in the PMHs?

Barrozo/Duarte et al. J. Am. Chem. Soc. 2015, 137, 9061 2828 Miton et al. in preparation Examples: Enzyme Catalysis

How does the reaction take place in solution?

‡

RS TS PS

Duarte et al. J. Am. Chem. Soc. 2016, accepted 2929 Duarte et al. J. Am. Chem. Soc. 2015, 137, 1081 Revisiting Fundamental Chemistry

‡ O O O O H H O P OR O P OR HO P O R HO P HO R O O H O O H O O OH O

O O O H O P OR HO P OR HO P HO R H O O O HO O O ‡ O O O O H H H O P OR O P OR O P O R HO P O R O H O H H O O O O O OH

30 Duarte et al. J. Am. Chem. Soc. 2015, 137, 1081 Examples: Enzyme Catalysis

M2+ M2+ O H O 2 O fGly57 C fGly57 C H OH O PO R 3 1 P BH O OR R1

M2+ M2+ O O HA O P O P fGly57 C O R O(H) fGly C O OR R 57 OH 1 R1 OH B

31 Barrozo/Duarte et al. J. Am. Chem. Soc. 2015, 137, 9061 Examples: Enzyme Catalysis

Coincidental pH rate proﬁles suggest same mechanism, same active site

• Mn2+: lewis-acid substantially reduces nucleophile pKa • pKE1: nucleophile • pKE2: general acid

Barrozo/Duarte et al. J. Am. Chem. Soc. 2015, 137, 9061 3232 Examples: Enzyme Catalysis

Phopshate Hydrolysis (sN2 type reaction)

K337 K337

H N H N H H O H O X=P,S O X O O X HO O FGl R 57 FGl57 R=O-, OCH CH , Ph R 2 3 H O H OH OH

NO2 NO2

33 Barrozo/Duarte et al. J. Am. Chem. Soc. 2015, 137, 9061 Examples: Enzyme Catalysis

- Q Software Package (Version 5.0.6)

- OPLS Force Field (Macromodel 9.1) /TIP3P III - Long-range Eﬀects: Local Reaction Field (LRF) approach II - Boundary: Surface Constrained All Atom Solvent Model I (SCAAS) I - EVB-Free Energy Perturbation / Umbrella Sampling Technique - 10 calculation/system (51 US windows/ 200ps each)

Warshel, A.; Weiss, R. M. J. Am. Chem. Soc. 1980, 102, 6218 34 Marelius, J. et al. J. Mol. Graph. Model. 1998, 16, 213 Examples: Enzyme Catalysis

I B.

] -1 15

[kcal ·mol [kcal

‡ ‡ G

∆ 5

0 WTWT Q13A N78A Y105AY105A T107A H218H218A K337K337A 35 Barrozo/Duarte et al. J. Am. Chem. Soc. 2015, 137, 9061 Enantio- and Regioselectivity: Epoxide Hydrolase - StEH1

36 Amrein et al. ACS Catalysis 2015, 5, 5702 Enantio- and Regioselectivity: Epoxide Hydrolase - StEH1

37 Amrein et al. ACS Catalysis 2015, 5, 5702 Enantio- and Regioselectivity: Epoxide Hydrolase - StEH1

38 Amrein et al. ACS Catalysis 2015, 5, 5702 Some References Diﬀerent Flavours of VB

Program Capabilities Website Comments Reference

MOLARIS- http://laetro.usc.edu/ Available for purchase from Warshel and EVB, FEP, AC XG software.html the USC coworkers

http://xray.bmc.uu.se/ Free for academic use. Åqvist and EVB, FEP, LIE Q ~aqwww/q/ Available upon request coworkers

Interface program that VB/MM Available upon request Shurki and communicates between VM/MM DE-VB/MM [email protected]. coworkers XMVB and MOLARIS

In house implementation in Voth and MS-EVB MS-EVB LAMMPS MD package coworkers

Tinker/ Glowacki and EVB In house implementation Charmm coworkers

Distributed Gaussian AMBER license is required Case and http://ambermd.org/ AMBER EVB for GPU version coworkers

40 Acknowledgements

Kamerlin/Åqvist Lab. Department of Cell and Molecular Biology, Uppsala

Paton Group. Department of Chemistry, Oxford

Newton International Fellowship, Royal Society

School of Chemistry, Edinburgh