Efficient computation of molecular response and excited state

properties

Filipp Furche

Institut fur Physikalische Chemie,

Universitat Karlsruhe

Bridging the gap

Working equations

Density-matrix based response theory I

• Basic variable: The non-interacting TDKS density matrix

γ(t, x, x′) =N∑

j=1

ϕj(t, x)ϕ∗j (t, x

′)

• Physical time-dependent density and current density accessible (cf. previous talks),

ρ(t, x) = γ(t, x, x),

j(t, x) = 1/2[

π(t, x) + π∗(t, x′)

]

γ(t, x, x′)∣

∣

x′=x

• Equations of motion:

i∂

∂tγ(t) = [HKS(t), γ(t)]

γ(t) = γ2(t)

Density-matrix based response theory II

• Periodic perturbation with coupling constant λ,

HKSλ(t) =1

2π2(t) + v

(0)ext(x) +

∫

dx′ ρλ(t, x′)

|r − r′|+ vxc[ρλ](t, x) − cxv

x,HF[γλ](t)

+ λ(

v(1)(ω, x)eiωt + v(1)(−ω, x)e−iωt)

• Response properties from order-by-order solution of the equations of motion for γλ(t)

• Advantages:

– Unitary invariant version of TDKS scheme

– Response from straightforward differentiation

– No response functions, no orbital derivatives, efficient implementation

– Straightforward matrix equivalent

– Efficient ground-state density matrix techniques transferable

F. F., J. Chem. Phys. 114 (2001), 5982.

Frequency depdendence

• Notation:

γ(n)(t) =1

n!

∂

∂λγλ(t)

∣

∣

∣

∣

λ=0

• Since the external perturbing potential is periodic with frquency ω, the n-th order

response has characteristic periodicity:

γ(0)(t)= γ(0)

γ(1)(t)= γ(1)(ω)eiωt + γ(1)(−ω)e−iωt

γ(2)(t)= γ(2)(ω, ω)e2iωt + γ(2)(ω,−ω) + γ(2)(−ω, ω) + γ(2)(−ω,−ω)e−2iωt

...

Zeroth order: Static KS

• Zeroth order idempotency constraint,

γ(0) = γ(0)γ(0),

equivalent to spectral representation

γ(0)(x, x′) =N∑

i=1

ϕi(x)ϕi(x′)

• Zeroth order EOM,

0 = [H(0)KS , γ(0)],

equivalent to static (hybrid) KS equations,

H(0)KSϕi(x) = ǫiϕi(x)

First order: Linear response I

• First order idempotency constraint:

γ(1) = γ(0)γ(1) + γ(1)γ(0)

• Static KS orbital basis:

γ(1)(x, x′) =∑

ia

(

Xiaϕa(x)ϕi(x′) + Yiaϕi(x)ϕa(x′)

)

– γ(1) ∈ L = Lvirt ⊗ Locc ⊕ Locc ⊗ Lvirt:

γ(1) =

(

X

Y

)

= |X, Y 〉

– Since γ(1)(t) = γ(1)†(t) we have

X(ω) = Y (−ω)

Y (ω) = X(−ω)

First order: Linear response II

• First order EOM:

ωγ(1) = [H(0)KS , γ(1)] + [H

(1)KS , γ(0)]

– Functional chain rule:

H(1)KS(ω) =

∫

dx′

(

1

|r− r′|+ fxc(ω, x, x′)

)

ρ(1)(ω, x′) − cxvx,HF[γ(1)](ω)

+ v(1)(ω, x)

– Adiabatic approximation:

fxc(ω, x, x′) = fxc(0, x, x′) =δ2Exc

δρ(x)δρ(x′)

First order: Linear response III

• EOM in static KS orbital basis:

(Λ − ω∆)|X, Y 〉 = −|P, Q〉

– Super-operators:

Λ =

(

A B

B A

)

, ∆ =

(

1 0

0 −1

)

– Orbital rotation Hessians:

(A + B)iajb = (ǫa − ǫi)δijδab + 2〈ab|ij〉 + 2fxciajb − cx[〈ab|ji〉 + 〈aj|bi〉]

(A − B)iajb = (ǫa − ǫi)δijδab + cx[〈ab|ji〉 − 〈aj|bi〉]

– External perturbation:

(P + Q)ia = 2〈i|v(1)(ω)|a〉, (P − Q)ia = 0

First order: Linear response IV

• Example: Polarizability αij(ω)

|P, Q〉 = |µj〉

αij(ω) = 〈µi|X, Y 〉 = 〈µi|(Λ − ω∆)−1|µj〉

• Hylleraas variational principle: Find stationary point of

F [X, Y ] = 〈X, Y |Λ − ω∆|X, Y 〉 + 〈X, Y |P, Q〉 + 〈P, Q|X, Y 〉

• First variation of F :

δF [X, Y ]

δ〈X, Y |= (Λ − ω∆)|X, Y 〉 + |P, Q〉 = 0

Higher order

• n-th order idempotency constraint: occ-occ and virt-virt parts of γ(n) are products of

lower-order quantities

• n-th order EOM: occ-virt and virt-occ part |X(n), Y (n)〉 satisfies

(Λ − ω∆)|X(n), Y (n)〉 = −|P (n−1), Q(n−1)〉,

where |P (n−1), Q(n−1)〉 contains lower-order quantities

• γ(n) determines (quasi-)energy to order 2n + 1 (Wigner’s rule)

Excited states I

• First order response of the TDKS density matrix

|X, Y 〉 = −(Λ − Ω∆)−1|P, Q〉

diverges if ω → Ωn.

• Determine poles and residues of (Λ − Ω∆)−1:

(Λ − Ωn∆)|Xn, Yn〉 = 0, 〈Xn, Yn|∆|Xn, Yn〉 = 1

– Ωn interacting excitation energies

– |Xn, Yn〉 yields interacting transition (current) densities or “collective density

modes”

– Examples: Oscillator and rotatory strength

f0n =2

3Ωn|〈µ|Xn, Yn〉|

2

R0n = Im〈µ|Xn, Yn〉 · 〈Xn, Yn|m〉

Excited states II

• Hylleraas variational principle: Find stationary point of

G[X, Y, Ω] = 〈X, Y |Λ|X, Y 〉 − Ω(〈X, Y |∆|X, Y 〉 − 1)

• First variation of G:

δG[X, Y, Ω]

δ〈X, Y |= (Λ − Ω∆)|X, Y 〉 = 0

∂G[X, Y, Ω]

∂Ω= 〈X, Y |∆|X, Y 〉 − 1

• Sum rules from spectral representation

(Λ − z∆)−1 =∑

n

(

1

Ωn − z|Xn, Yn〉〈Xn, Yn| +

1

Ωn + z|Yn, Xn〉〈Yn, Xn|

)

Summary

• TDDFT is based on a mapping of the interacting problem on the non-interacting

TDKS system (cf. previous talks)

• Response properties of the interacting system can be extracted from the TDKS

response

• The central equation of TDKS response theory is (TDKS response equation)

(Λ − ω∆)|X, Y 〉 = −|P, Q〉

• Excitation energies and transition moments are accessible from the poles of the TDKS

density matrix response

• The central equation of TDKS excited state theory is (TDKS eigenvalue problem,

“Casida’s equations”)

(Λ − Ω∆)|X, Y 〉 = 0, 〈X, Y |∆|X, Y 〉 = 1

Implementation

Discretization

• Generate a finite number of MOs by the LCAO-MO expansion,

φpσ(r) =∑

µ

Cµpσχµ(r)

• TDKS response/eigenvalue problem becomes finite dimensional linear equation

system/eigenvalue problem → matrix algebra

• Basis set error, can be controlled by extrapolation

• Contracted Cartesian Gaussians, the better choice for molecules:

χµ(r) =∑

p

cpxiyjzke−ζp(r−Rµ)2

– Gaussian product theorem → multi-center integrals

– Hermite recursion: Higher l quantum numbers from derivatives with respect to Rµ

Solution by elimination techniques

(A + B)iaσjbσ′ = (ǫaσ − ǫiσ)δijδabδσσ′+2(iaσ|jbσ′) + 2fxciaσjbσ′

−cxδσσ′ [(jaσ|ibσ) + (abσ|ijσ)]

(A − B)iaσjbσ′ = (ǫaσ − ǫiσ)δijδabδσσ′

+cxδσσ′ [(jaσ|ibσ) − (abσ|ijσ)]

The dimension of L scales as Nocc · Nvirt!

Naive approach:

• Calculate all integrals and store them → CPU ∼ N5, I/O ∼ N4

• Use elimination techniques to solve the linear equation system / eigenvalue problem →

CPU ∼ N6

Prohibitive for more than 10 atoms

The end of TDDFT (in chemistry)!

Iterative Methods

• In most applications, only a small number of perturbations / excited states is of

interest

• Minimize the functionals

F [X, Y ] = 〈X, Y |Λ − ω∆|X, Y 〉 + 〈X, Y |P, Q〉 + 〈P, Q|X, Y 〉,

G[X, Y, Ω] = 〈X, Y |Λ|X, Y 〉 − Ω(〈X, Y |∆|X, Y 〉 − 1)

on an iteratively expanded subspace of L

• Requires small number of matrix-vector products Λ|X, Y 〉

• Using these techniques A and B never need to be set up and stored!

• First proposed by Hestenes, Stiefel, and Lanzcos, useful preconditioning by Davidson

Iterative subspace methods

Chose orthonormal

start vector space S

Compute projection

of Λ on S

Minimize F or G on S

(Ritz step)

→ |ξ, η〉 ∈ S, Ω

Gradient |r, s〉 =

(Λ − ω∆)|ξ, η〉 + |P, Q〉

or |r, s〉 = (Λ − Ω∆)|ξ, η〉

‖|r, s〉‖

< thr ?

yes

no

Include orthonormalized

and preconditioned |r, s〉

as new basis vector in S

Converged

Integral direct iterative algorithms I

• Usually less than 10 iterations required for residual norm ≤ 10−5

• A posteriori error bounds available

• Integral direct algorithm:

– Transform vectors, not integrals

– Compute non-vanishing integrals “on the fly”, i.e., discard them after use →

pre-screening

• Time-determining step: Matrix-vector products

|U, V 〉 = Λ|X, Y 〉,

best calculated as

(U ± V ) = (A ± B)(X ± Y )

Integral direct iterative algorithms II

• Transformation MO-AO

(X ± Y )µνσ =1

2

∑

ia

(X ± Y )iaσ(CµiσCνaσ ± CµaσCνiσ)

• Matrix-vector multiplication (AO basis)

(U + V )µνσ =∑

κλσ′

(

2(µν|κλ) + 2fxcµνσκλσ′

− cxδσσ′ [(µκ|νλ) + (µλ|νκ)])

(X + Y )κλσ′

(U − V )µνσ =∑

κλσ′

cxδσσ′ [(µκ|νλ) − (µλ|νκ)](X − Y )κλσ′

• Transformation AO-MO

(U ± V )iaσ →1

2

∑

µν

(U ± V )µνσ(CµiσCνaσ ± CµaσCνiσ)

Two-electron integrals (µν|κλ)

Well-established techniques from ground-state quantum chemistry:

• A priori bound from Cauchy-Schwarz inequality,

|(µν|κλ)| ≤ (µν|µν)1/2(κλ|κλ)1/2

Result: O(N4) → O(N2)

• Analytical expression for (ss|ss), higher l by analytical recursion (Obara-Saika)

• Auxiliary expansion (RI) techniques:

(µν|κλ) →∑

pq

(µν|p)(p|q)−1(q|κλ)

– Prefactor reduced by 10 − 100

– Efficient for Coulomb only

– More in Dmitrij Rappoport’s talk

Exchange-correlation contributions

(U + V )µνσ →∑

κλσ′

fxcµνσκλσ′(X + Y )κλσ′

Example: ALDA,

Exc[ρ] =

∫

d3r f(ρα(r), ρβ(r))

1. Evaluate “density response” on the grid O(N3) → O(N)

ρ(1)(ri) =∑

µν

χµ(ri)χν(ri)(X + Y )µνσ

2. Perform quadrature O(N3) → O(N)

(U + V )µνσ →∑

i

wiχµ(ri)χν(ri)∑

σ′

∂2f

∂ρσ(ri)∂ρσ′(ri)ρ(1)σ′ (ri)

Equivalent to computation of the first order response of the xc potential!

Turbomole implementation escf

System Sym. Method NBF pa CPU Rb

Tris(Alanine)-CoIII C3 B3LYP 386 100 12:04 B

Cu-phthalocyaninc D4h B3LYP 706 90 40:24 B

Tetrathia-[7]helicene C2 B3LYP 482 50 30:13 B

Fullerene C540 Ih BP/RI 8100 3 19:17 B

“Cd10Se16”d T BP/RI 2804 300 128:04 A

Vancomycin C1 BP/RI 1294 100 46:08 B

Methylcobalamin C1 BP/RI 1600 100 62:18 A

anumber of excited statesbplatform used. A: 1.2 GHz Athlon PC, B: 440 MHz HP J5000copen shelldCd10Se4(SePh)12(P

nPr3)4

Turbomole V5-4

Present limit (single processor): 200-1000 atoms

Excited state properties

• Problem: Linear response theory supplies excitation energies and transition moments,

but no excited state wavefunction!

• Solution: Define excited state properties via the response of the excited state energy

Ees[v].

• Examples:

– Excited state dipole moments: ∂Ees[v]∂ǫ

∣

∣

∣

ǫ=0

– Excited state density δEes[v]δvext(r)

∣

∣

∣

vext(r)=0

– Excited state gradients ∂Ees[v]∂Rµ

∣

∣

∣

Rµ=0

• WANTED: Fully variational expression for the excited state energy

Lagrangian of the excited state energy

• Define Lagrangian

L[X, Y, Ω, C, Z, W ] = EGS + G[X, Y, Ω] +∑

ia

ZiaH(0)KS ia −

∑

pq

Wpq(Spq − δpq)

– EGS: Ground state energy functional

– S: Overlap matrix

– Z, W : Additional Lagrange multipliers

• Result:The minimum of L with respect to all parameters is the total energy

of the first excited state.

F. F., R. Ahlrichs, J. Chem. Phys. 117 (2002), 7433.

Stationarity conditions

(1)∂L

∂Zia= Fia = 0,

∂L

∂Wpq= Spq − δpq = 0

Ground state KS equations → Molecular orbital coefficients C

(2)∂L

∂〈X, Y |= (Λ − Ω∆)|X, Y 〉 = 0,

∂L

∂Ω= 〈X, Y |∆|X, Y 〉 − 1 = 0

TDKS eigenvalue problem (Casida’s equations) → Excitation energy Ω, excitation vector

|X, Y 〉

(3)∂L

∂C= 0

Z vector equation (A + B)Z = −R → relaxed excited state density matrix P , energy

weighted relaxed density matrix W

Excited state properties

Lξ =∑

µνσ

hξµνPµνσ −

∑

µνσ

SξµνWµνσ +

∑

µνκλσσ′

(µν|κλ)ξΓµνσκλσ′

+∑

µνσ

V xc (ξ)µνσ Pµνσ +

∑

µνκλσσ′

fxc (ξ)µνσκλσ′(X + Y )µνσ(X + Y )κλσ′

• General expression for integral-direct evaluation of excited state first-order properties

• Almost identical to ground state expression → same O(N)/O(N2) scaling

• Variational stability of L → No MO coefficient derivatives Cξµpσ

• The cost for computing excited state gradients analytically is independent of the

number of nuclear degrees of freedom

Turbomole implementation: egrad

GS energy 0:33 h

ES energy+grad 1:44 h

opt. cycles 13

Chlorophyll a, 2 1A

Turbomole V5-7, RIDFT/BP-86/SV(P), 1.7 GHz Athlon PC

Excited state geometry optimization is not significantly more expensive than ground state

geometry optimization

D. Rappoport, F. F., J. Chem. Phys. 122 (2005), 064105.

Summary: Implementation I

• Basis set expansion reduces TDKS response and eigenvalue problems to

finite-dimensional matrix algebra

• Iterative subspace methods:

– Only two-index quantities (vectors) need to be stored and handeled, no four-index

MO integrals

– Time-determining steps: O(N2)–O(N) scaling, very similar to ground-state Fock

matrix construction

– Significant savings by simultaneous processing of several perturbations/states

– More efficient than real-time propagation for linear response

• Solving the TDKS response equations / the TDKS eigenvalue problem is less

expensive than solving the ground state KS equations

Summary: Implementation II

• Excited state properties are best calculated as analytical derivatives of the excited

state Lagrangian

• Analytical derivatives yield exact response of the excited state energy in any basis set

• The cost of analyical excited state gradients does not scale with the number of nuclear

degrees of freedom

• Large systems are the domain of TDDFT in quantum chemistry → implementation

matters

• Present limit (single processor): 200-1000 atoms

Further reading

[1] Time-dependent density functional theory in quantum chemistry. F. F. and K. Burke,

in Annual Reports in Computational Chemistry 1, edited by D. C. Spellmeyer, Elsevier,

Amsterdam, 2005, p. 19.

[2] Density functional methods for excited states: equilibrium structure and electronic

spectra. F. F. and D. Rappoport, In Computational Photochemistry, edited by M.

Olivucci, Elsevier, Amsterdam, 2005, p. 93.

[3] Time-dependent density functional theory, edited by M. Marques, C. A. Ullrich, F.

Nogueira, A. Rubio, K. Burke, and E. K. U. Gross, Springer, Berlin, 2006.

Acknowledgments

• Dmitrij Rappoport

• Quantum chemistry groups (Karlsruhe)

• Reinhart Ahlrichs (Karlsruhe)

• Kieron Burke (Irvine)

www.turbomole.com