Nonadiabatic Couplings and Conical Intersections ...€¦ · Geometry...

Geometry Optimization Gradients Nonadiabatic Couplings

Nonadiabatic Couplings and Conical Intersections:Algorithmic Details

Felix Plasser

Institute for Theoretical Chemistry, University of Vienna

COLUMBUS in ChinaTianjin, October 10–14, 2016

F. Plasser NAC and Con. Ints 1 / 35


Potential Energy Surfaces

I How to find minima?I How to find conical intersections?→ Geometry optimization



Minima

I Task: find a local minimum of the PES EI(R)with respect to the nuclear coordinates R = R1, . . . , Rn

/ We do not know the functional form of EI(R), We can compute EI(R) at individual geometries, We can compute the energy gradient 5EI(R)



Minima

Newton-Raphson MethodI Find a root of a one-dimensional function f(x)

Taylor expansion

f(x) = f(x0) + (x− x0)f ′(x0)

I Try to find point x1 where f(x1) = 0

x1 = x0 −f(x0)

f ′(x0)

I Iterate



Minima

Newton-Raphson MethodI Iterative procedure



Minima

Newton-Raphson MethodI Find a stationary point of a one-dimensional function f(x)

Stationary point

f ′(x) = 0

I Apply Newton-Raphson to the derivative

x1 = x0 −f ′(x0)

f ′′(x0)

I Iterate



Minima

Newton-Raphson MethodI Find a stationary point of a multi-dimensional function EI(R)

Gradient vector

g(R) = 5EI(R)

gi(R) =∂

∂RiEI(R)

Talyer expansion of the gradient

g(R) = g(R0) + H · (R−R0)

Hij =∂2

∂Ri∂RjEI(R)



Minima

Talyer expansion of the gradient

g(R) = g(R0) + H · (R−R0)

Hij =∂2

∂Ri∂RjEI(R)

I Set the gradient to zero

R1 = R0 −H−1g(R)

I Iterate



Minima

Quasi-Newton-Raphson Method

R1 = R0 −H−1g(R)

I Estimate H−1

I Initial guess: diagonal Hessian with respect to internal coordinatesintcfl file

I Hessian update



Minima

Direct inversion in the iterative subspace (DIIS)I Consider all previous geometries in the optimizationI gdiis program



MXS optimization

MXS optimization

Task EI(R)+EJ(R)2 → minCondition EI(R) = EJ(R)

/ The surfaces are not differentiable at the conical intersection, We can compute the coupling vectorsI Idea: separate the intersection and branching spacesI Optimize the energy in the intersection spaceI Minimize the gap in the branching space



MXS Optimization

Several methods availableI Gradient projectionI Lagrange NewtonI Penalty function



MXS Optimization

Gradient projectionI Projected gradient in the intersection space

Projected gradient

gp =(1− gIJgIJT − hIJhIJT

)5 EJ

gIJ Gradient difference vectorhIJ Nonadiabatic coupling vector

I Minimization of the gap in the branching space- Follow gIJ

1 Beapark, Robb, and Schlegel 1994, .F. Plasser NAC and Con. Ints 15 / 35


MXS Optimization

Lagrange NewtonI Introduce Lagrange multipliers for the degeneracy

Lagrangian

L(R, λ1, λ2) = EI(R) + λ1∆EIJ(R) + λ2HIJ

I Optimize the Lagrangian using Newton-Raphson/GDIISI In COLUMBUS

1 M. Dallos et al. J. Chem. Phys. 2004, 120, 7330.F. Plasser NAC and Con. Ints 16 / 35


MXS Optimization

Penalty function

f(R) =EI + EJ

2c1c

22 ln

(1 +

EI − EJc2

)2I Minimize a penalty function- Optimize the energy- Minimize the gapI Optimization without nonadiabatic coupling vectors

1 Ciminelli, Granucci, and Persico 2004, .F. Plasser NAC and Con. Ints 17 / 35


Gradients

Energy expectation value

E = 〈Ψ| Ĥ |Ψ〉E(R) = 〈Ψ(k (R))| Ĥ(R) |Ψ(k (R))〉

R Nuclear geometryk(R) Wavefunction parameters (e.g. orbital and CI coefficients)



Gradients

Energy expectation value

E(R) = 〈Ψ(k (R))| Ĥ(R) |Ψ(k (R))〉

Energy gradient

∂

∂RxE(R) ≡ Ex = 〈Ψ| Ĥx |Ψ〉+ 2

∑i

〈Ψ| Ĥ∣∣∣∣ ∂∂ki Ψ

〉kxi

Rx One nuclear coordinateEx Derivative of the energy with respect to Rxkxi Derivative of wavefunction parameter ki with respect to Rx

Chain rule!



Gradients

Variational methods

Energy gradient

Ex = 〈Ψ| Ĥx |Ψ〉+ 2∑i

〈Ψ| Ĥ∣∣∣∣ ∂∂ki Ψ

〉kxi

Variationally optimized wavefunction

0 =∂

∂ki〈Ψ| Ĥ |Ψ〉 = 2 〈Ψ| Ĥ

∣∣∣∣ ∂∂ki Ψ〉

I Generalized Hellman-Feynman theorem

Energy gradient

Ex = 〈Ψ| Ĥx |Ψ〉



Gradients

Variational methodsI State-specific MCSCF

, VariationalI State-averaged MCSCF

/ Orbitals not variationalOrbitals are optimized for the average energy and not for the individual states

I MR-CI/ Orbitals not variational



Gradient

State-specific MCSCF

Energy gradient

Ex = 〈Ψ| Ĥx |Ψ〉

Practical stepsI Compute the 1- and 2-particle density matrices (DM)

mcscf.xI Transform the DMs to the AO basis

tran.xI Compute the AO integral derivatives and contract them with the DMs

dalton.x “abacus”



Gradients

Non-variational methods

Energy gradient

Ex = 〈Ψ| Ĥx |Ψ〉+ 2∑i

〈Ψ| Ĥ∣∣∣∣ ∂∂ki Ψ

〉kxi

= 〈Ψ| Ĥx |Ψ〉+∑i

fikxi

= 〈Ψ| Ĥx |Ψ〉+ f · kx

f Wavefunction gradientHow does the energy change if I change the parameters? – easy

kx Geometric parameter derivativesHow do the parameters change if I change the geometry? – difficult



Gradients

We know how the reference wavefunction was constructed!

Reference wavefunction

fref(R,k(R)) = 0

fref Wavefunction gradient of the reference wavefunction, e.g. MCSCFVanishes since the reference wavefunction was optimized

Geometric derivative

fxref +∑j

∂

∂kjfref · kx = 0

fxref + Gkx = 0

G Wavefunction Hessian of the reference wavefunction



Gradients

Geometric derivative

fxref + Gkx = 0

kx = −G−1fxref

Energy gradient

Ex = 〈Ψ| Ĥx |Ψ〉+ f · kx

Ex = 〈Ψ| Ĥx |Ψ〉 − fTG−1fxref



Gradients

Energy gradient

Ex = 〈Ψ| Ĥx |Ψ〉 − fTG−1fxref

I Coupled perturbed MCSCF equationsI Trick: Precompute fTG−1 as an equation system1

I Z-vector equation

Z-vector equation

λT = fTG−1

λTG = fT

I Independent of the geometric derivativeI cigrd.x program

1 N. C. Handy, H. F. Schaefer III J. Chem. Phys. 1984, 81, 5031.F. Plasser NAC and Con. Ints 27 / 35


Gradients

I Convert the result into effective density matricesI Contract with the AO derivative integrals, as before



Gradient

State-averaged MCSCF, MR-CIPractical steps

I Compute the 1- and 2-particle density matrices (DM)mcscf.x, (p)ciudg.x

I Solve the coupled perturbed equationscigrd.x

I Transform the effective DMs to the AO basistran.x

I Compute the AO integral derivatives and contract them with the DMsdalton.x “abacus”



Gradients

In reality, it is more complicated ...

I Two types of wavefunction parameters− CI-coefficients – variationally optimized for the individual states− MO-coefficientsI Redundant parameters- Orbital resolution can affect MR-CI energies and gradients

1 H. Lischka, M. Dallos, R. Shepard Mol. Phys. 2002, 100, 1647.F. Plasser NAC and Con. Ints 30 / 35


Nonadiabatic Couplings

Nonadiabatic coupling

h12 = 〈Ψ1|5Ψ2〉hx12 = 〈Ψ1|Ψx2〉

I Measures changes in the wavefunctionI Is it related to the gradient?




Schrödinger Equation

Ĥ |Ψ2〉 = E2 |Ψ2〉Ĥx |Ψ2〉+ Ĥ |Ψx2〉 = Ex2 |Ψ2〉+ E2 |Ψx2〉

〈Ψ1| Ĥx |Ψ2〉+ 〈Ψ1| Ĥ |Ψx2〉 = 〈Ψ1|Ex2 |Ψ2〉+ 〈Ψ1|E2 |Ψx2〉〈Ψ1| Ĥx |Ψ2〉+ E1 〈Ψ1|Ψx2〉 = E2 〈Ψ1|Ψx2〉


hx12 = 〈Ψ1|Ψx2〉 =〈Ψ1| Ĥx |Ψ2〉E2 − E1





hx12 = 〈Ψ1|Ψx2〉 =〈Ψ1| Ĥx |Ψ2〉E2 − E1

I Wavefunction derivative converted into Hamiltonian derivativeI Equation similar to gradient→ Use similar methodology




Nonadiabatic couplings vs. gradients

I Use transition density matrices instead of density matricesI Formalism somewhat more involved, two terms- CI coefficient derivative “DCI”- CSF coefficient derivative “DCSF”


Geometry OptimizationGradientsNonadiabatic Couplings

Nonadiabatic Couplings and Conical Intersections ...€¦ · Geometry...

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