Calculation of molecular properties by response theory

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Calculation of molecular properties by response theory

Julien Toulouse Laboratoire de Chimie Th´eorique

Sorbonne Universit´ e and CNRS, Paris, France

Institut Universitaire de France

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Books:

Roy McWeeny,Methods of Molecular Quantum Mechanics, Academic Press, 1992.

Stephan Sauer,Molecular Electromagnetism: A Computational Chemistry, Oxford Graduate Texts, 2011.

Patrick Norman, Kenneth Ruud, Trond Saue,Principles and Practices of Molecular Properties: Theory, Modeling and Simulations, Wiley, 2018.

Review articles:

M. Casida,Time-dependent density-functional theory for molecules and molecular solids, Journal of Molecular Structure: THEOCHEM 914, 3, 2009.

T. Helgaker, S. Coriani, P. Jørgensen, K. Kristensen, J. Olsen, K. Ruud,Recent Advances in Wave Function-Based Methods of Molecular-Property Calculations, Chemical Review 112, 543, 2012.

My lecture notes:

www.lct.jussieu.fr/pagesperso/toulouse/enseignement/molecular_properties.pdf

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1 Static molecular properties Definition and examples General static response theory Linear response HF and DFT

2 Dynamic molecular properties Definition from the quasi-energy

Polarizability from dynamic response theory Linear response TDHF and TDDFT

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Consider the isolatedmolecular HamiltonianHˆ0= ˆT+ ˆVne+ ˆWee and apply astatic perturbationVˆ(x)

H(x) = ˆˆ H₀+ ˆV(x)

wherex represents the strength of the perturbation. The perturbation vanishes for x = 0, i.e. ˆV(x= 0) = 0

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Consider the isolatedmolecular HamiltonianHˆ0= ˆT+ ˆVne+ ˆWee and apply astatic perturbationVˆ(x)

H(x) = ˆˆ H₀+ ˆV(x)

wherex represents the strength of the perturbation. The perturbation vanishes for x = 0, i.e. ˆV(x= 0) = 0

Theperturbed energyfor the perturbed state considered Ψ(x) (usually the ground state) can be expanded with respect tox

E(x) = hΨ(x)|H(xˆ )|Ψ(x)i

hΨ(x)|Ψ(x)i =E⁽⁰⁾+E⁽¹⁾x+1

2E⁽²⁾x²+· · ·

whereE⁽⁰⁾ is the unperturbed energy and theenergy derivatives with respect tox E⁽¹⁾=dE

dx

x=0

; E⁽²⁾= d²E dx²

x=0

; etc...

are calledstatic (or time-independent) molecular properties

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Geometrical derivatives: perturbation strengthsx are the variations of the nuclei positionsRi−R_i⁰(in the Born-Oppenheimer approximation). For example, the molecular gradient and Hessianare

∂E

∂Ri

and ∂²E

∂Ri∂Rj

which are needed for geometry optimization and calculating harmonic vibrational frequencies

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∂E

∂Ri

and ∂²E

∂Ri∂Rj

Electric properties: perturbation strengthsx are the components of the external static electric fieldEi. For example, theelectric dipole moment and polarizabilityare

∂E

∂Ei

E=0

=−µi and ∂²E

∂Ei∂Ej

E=0

=−αi,j

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∂E

∂Ri

and ∂²E

∂Ri∂Rj

Electric properties: perturbation strengthsx are the components of the external static electric fieldEi. For example, theelectric dipole moment and polarizabilityare

∂E

∂Ei

E=0

=−µi and ∂²E

∂Ei∂Ej

E=0

=−αi,j

NMR magnetic properties: perturbation strengthsx are the nuclear magnetic dipole momentsmaand the external magnetic fieldB. For example, theNMR shielding

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There are two ways of calculating energy derivatives:

Numerical differentiation: the derivatives are calculated by finite differences or by polynomial fitting

It is simple to implement.

The numerical precision is limited.

The computational efficiency is low.

It is not general. Usually, only real-valued and static perturbations can be done.

Analytical differentiation: the derivatives are calculated explicitly from the analytical expressions

It is difficult to implement.

The precision is higher.

The computational efficiency is higher.

It is more general. In particular, time-dependent perturbations are possible.

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For “exact” wave functions (full configuration interaction in a basis), we can use straightforwardperturbation theoryto find the expression of the energy derivatives

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For “exact” wave functions (full configuration interaction in a basis), we can use straightforwardperturbation theoryto find the expression of the energy derivatives For example, for a perturbation linear inx,

H(xˆ ) = ˆH₀+xVˆ thefirst-order energy derivativeis

E⁽¹⁾=hΨ0|Vˆ|Ψ0i and thesecond-order energy derivativeis

E⁽²⁾=−2X

n6=0

hΨ0|Vˆ|ΨnihΨn|Vˆ|Ψ0i En− E0

where{Ψn}and{En}are the eigenstates and eigenvalues of the unperturbed

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For “exact” wave functions (full configuration interaction in a basis), we can use straightforwardperturbation theoryto find the expression of the energy derivatives For example, for a perturbation linear inx,

H(xˆ ) = ˆH₀+xVˆ thefirst-order energy derivativeis

E⁽¹⁾=hΨ0|Vˆ|Ψ0i and thesecond-order energy derivativeis

E⁽²⁾=−2X

n6=0

hΨ0|Vˆ|ΨnihΨn|Vˆ|Ψ0i En− E0

where{Ψn}and{En}are the eigenstates and eigenvalues of the unperturbed Hamiltonian ˆH0

However, these expressions cannot be applied for methods like HF or KS DFT

=⇒we need a moregeneral response theory

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In an electronic-structure method, theenergyE(x) is obtained by optimizing parameters p= (p1,p2, ....) in anenergy functionE(x,p) for each fixed value ofx. The energy is thus

E(x) =E(x,p^o(x)) wherep^o(x) are the optimal values of the parameters Note that the optimization is not necessarily variational!

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In an electronic-structure method, theenergyE(x) is obtained by optimizing parameters p= (p1,p2, ....) in anenergy functionE(x,p) for each fixed value ofx. The energy is thus

E(x) =E(x,p^o(x)) wherep^o(x) are the optimal values of the parameters Note that the optimization is not necessarily variational!

Examples:

HF and KS DFT:pare the orbital parameters(variational)

MCSCF:pare the coefficients of the configurations(variational)and the orbital parameters(variational)

CI:pare the coefficients of the configurations(variational)and the orbital parameters(non variational)

CC:pare the cluster amplitudes(non variational)and the orbital parameters(non variational)

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Thefirst-order derivativeofE(x) with respect tox is dE(x)

dx =∂E(x,p^o)

∂x +X

i

∂E(x,p)

∂pi

_p=p_o

∂p^o_i

∂x

| {z }

explicit dependence onx

| {z }

implicit dependence onx

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dx =∂E(x,p^o)

∂x +X

i

∂E(x,p)

∂pi

_p=p_o

∂p^o_i

∂x

| {z }

∂p^oi/∂x is called thelinear-response vectorand contains information about how the wave function changes when the system is perturbed. It is not straightforward to calculate it since we do not know explicitly the dependence ofp_i^oonx.

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dx =∂E(x,p^o)

∂x +X

i

∂E(x,p)

∂pi

_p=p_o

∂p^o_i

∂x

| {z }

If all the parameters arevariational, the zero electronic gradient condition simplifies the derivative

∂E(x,p)

∂pi

p=p^o

= 0 =⇒ dE(x)

dx =∂E(x,p^o)

∂x i.e., we do not need the linear-response vector

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dx =∂E(x,p^o)

∂x +X

i

∂E(x,p)

∂pi

_p=p_o

∂p^o_i

∂x

| {z }

If all the parameters arevariational, the zero electronic gradient condition simplifies the derivative

∂E(x,p)

∂pi

p=p^o

= 0 =⇒ dE(x)

dx =∂E(x,p^o)

∂x i.e., we do not need the linear-response vector

If all the parameters arevariational, the second-order derivative ofE(x) is then d²E(x)

dx² =∂²E(x,p^o)

∂x² +X

i

∂²E(x,p)

∂x∂pi

p=p^o

∂p^o_i

∂x We need now the linear-response vector∂pi^o/∂x

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To obtain the linear-response vector∂p^oi/∂x when all the parameters are variational, we start from thestationary conditionwhich is true for allx

∀x, ∂E(x,p)

∂pi

p=p^o

= 0

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∀x, ∂E(x,p)

∂pi

p=p^o

= 0

We can thus take the derivative of this equation with respect tox

∂²E(x,p)

∂x∂pi

p=p^o

+X

j

∂²E(x,p)

∂pi∂pj

p=p^o

∂pj^o

∂x = 0 or

X

j

∂²E(x,p)

∂pi∂pj

p=p^o

∂p^o_j

∂x =−∂²E(x,p)

∂x∂pi

p=p^o

which are thelinear-response equations

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∀x, ∂E(x,p)

∂pi

p=p^o

= 0

We can thus take the derivative of this equation with respect tox

∂²E(x,p)

∂x∂pi

p=p^o

+X

j

∂²E(x,p)

∂pi∂pj

p=p^o

∂pj^o

∂x = 0 or

X

j

∂²E(x,p)

∂pi∂pj

p=p^o

∂p^o_j

∂x =−∂²E(x,p)

∂x∂pi

p=p^o

which are thelinear-response equations

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The HF single-determinant wave function is parametrized by orbital rotation parametersκ

|Φ(κ)i=e^κ(^ˆ^κ⁾|Φi

wheree^ˆ^κ(^κ⁾performs orbital rotations and is written with the orbital excitation operator ˆ

κ(κ) = Xocc

a

Xvir r

κarˆcr^†ˆca−κ^∗arcˆa^†ˆcr

For the static case, we can take real-valued orbital rotation parameters, i.e. κar=κ^∗ar

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κ(κ) = Xocc

a

Xvir r

Without perturbation, theHF energy functionis E_HF(κ) =hΦ(κ)|Hˆ0|Φ(κ)i

hΦ(κ)|Φ(κ)i

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κ(κ) = Xocc

a

Xvir r

Without perturbation, theHF energy functionis E_HF(κ) =hΦ(κ)|Hˆ0|Φ(κ)i

hΦ(κ)|Φ(κ)i

For linear response theory, we need first to calculate theHF electronic Hessian

∂²EHF(κ)

∂κar∂κbs

κ=0

= 2 (Aar,bs+Bar,bs)

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We consider the perturbation by a static electric fieldE. Theperturbed HF energyis E_HF(E,κ) =hΦ(κ)|Hˆ₀−µˆ·E|Φ(κ)i

hΦ(κ)|Φ(κ)i

where ˆµis the dipole moment operator

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hΦ(κ)|Φ(κ)i

where ˆµis the dipole moment operator TheCPHF electric dipole polarizabilityis

α^CPHFi,j =−∂²EHF

∂Ei∂Ej

_E=0=−

"

∂²E_HF

∂Ei∂Ej

+ Xocc

a

Xvir r

∂²E_HF

∂Ei∂κar

κ=0

∂κar

∂Ej

#

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hΦ(κ)|Φ(κ)i

∂Ei∂Ej

_E=0=−

"

∂²E_HF

∂Ei∂Ej

+ Xocc

a

Xvir r

∂²E_HF

∂Ei∂κar

κ=0

∂κar

∂Ej

#

Since the explicit dependence ofEHF(E,κ) onEis linear, we have∂²EHF/∂Ei∂Ej= 0

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hΦ(κ)|Φ(κ)i

∂Ei∂Ej

_E=0=−

"

∂²E_HF

∂Ei∂Ej

+ Xocc

a

Xvir r

∂²E_HF

∂Ei∂κar

κ=0

∂κar

∂Ej

#

Since the explicit dependence ofEHF(E,κ) onEis linear, we have∂²EHF/∂Ei∂Ej= 0 TheHF perturbed electronic gradientis

∂²E

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TheCPHF polarizabilityis thus α^CPHFi,j =

Xocc a

Xvir r

2hr|ˆµi|ai∂κar

∂Ej

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Xocc a

Xvir r

2hr|ˆµi|ai∂κar

∂Ej

The linear-response vector∂κar/∂Ejis obtained from theHF linear-response equations Xocc

b

Xvir s

(Aar,bs+Bar,bs)∂κbs

∂Ej

=hr|ˆµj|ai

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Xocc a

Xvir r

2hr|ˆµi|ai∂κar

∂Ej

The linear-response vector∂κar/∂Ejis obtained from theHF linear-response equations Xocc

b

Xvir s

(Aar,bs+Bar,bs)∂κbs

∂Ej

=hr|ˆµj|ai

In vector/matrix notations, we obtain finally

α^CPHFi,j = 2µ^T_i (A+B)⁻¹µ_j

whereµ_i is the vector of componentsµi,ar=hr|ˆµi|aiandA+Bis the matrix of elementsAar,bs+Bar,bs

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TheKS-DFT energy functionis

EKS-DFT(κ) = hΦ(κ)|Tˆ + ˆV_ne|Φ(κ)i

hΦ(κ)|Φ(κ)i +EH[ρ_Φ(κ)] +Exc[ρ_Φ(κ)] whereEH[ρ] is the Hartree density functional andExc[ρ] is the exchange-correlation density functional

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TheKS-DFT electronic Hessianhas a similar expression as in HF

∂²E_KS-DFT(κ)

∂κar∂κbs

κ=0

= 2 (Aar,bs+Bar,bs) where

exchange-correlation kernel fxc(r1,r2) =δ²Exc[ρ]/δρ(r1)δρ(r2)

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TheKS-DFT electronic Hessianhas a similar expression as in HF

∂²E_KS-DFT(κ)

∂κar∂κbs

κ=0

= 2 (Aar,bs+Bar,bs) where

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If we neglect electron-electron interaction, we haveAar,bs= (εr−εa)δabδrs and

Bar,bs= 0, and we obtainuncoupled Hartree-Fock (UCHF)oruncoupled Kohn-Sham

(UCKS)polarizabilities

α^UCHF/UCKS_i,j = 2 Xocc

a

Xvir r

hr|ˆµi|aiha|ˆµj|ri εr−εa

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If we neglect electron-electron interaction, we haveAar,bs= (εr−εa)δabδrs and

Bar,bs= 0, and we obtainuncoupled Hartree-Fock (UCHF)oruncoupled Kohn-Sham

(UCKS)polarizabilities

α^UCHF/UCKS_i,j = 2 Xocc

a

Xvir r

hr|ˆµi|aiha|ˆµj|ri εr−εa

Spherically averaged polarizabilities of atoms at the UCHF/CPHF and UCKS/CPKS levels (LDA functional, uncontracted d-aug-cc-pCV5Z basis sets):

UCHF CPHF UCKS CPKS Estimated exact

He 1.00 1.32 1.81 1.66 1.38

Ne 1.98 2.38 3.48 3.05 2.67

Ar 10.1 10.8 18.0 12.0 11.1

Kr 15.9 16.5 27.8 18.0 16.8

Be 30.6 45.6 80.6 43.8 37.8

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Time-dependent Schr¨odinger equation and quasi-energy

We consider now aperiodic time-dependent perturbation H(t) = ˆˆ H0+x⁺V eˆ ^−iωt+x⁻Vˆ^†e^+iωt

wherex⁺ andx⁻represent the perturbation strengths (ultimatelyx⁺=x⁻)

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Time-dependent Schr¨odinger equation and quasi-energy

The associated wave function|Ψ(t)i¯ satisfies the time-dependent Schr¨odinger equation

H(t)ˆ −i∂

∂t

|Ψ(t)i¯ = 0

After extracting a phase factor,|Ψ(t)i¯ =e^−iF(t)|Ψ(t)i, it can be reformulated as

H(t)ˆ −i∂

∂t

|Ψ(t)i= ˙F(t)|Ψ(t)i (1) where ˙F(t) = dF/dt is thetime-dependent quasi-energy.

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Time-dependent Schr¨odinger equation and quasi-energy

The associated wave function|Ψ(t)i¯ satisfies the time-dependent Schr¨odinger equation

H(t)ˆ −i∂

∂t

|Ψ(t)i¯ = 0

After extracting a phase factor,|Ψ(t)i¯ =e^−iF(t)|Ψ(t)i, it can be reformulated as

H(t)ˆ −i∂

∂t

|Ψ(t)i= ˙F(t)|Ψ(t)i (1) where ˙F(t) = dF/dt is thetime-dependent quasi-energy.

Thequasi-energyQis defined as the time average of ˙F(t) over a periodT = 2π/ω:

Q= 1 T

ZT 0

dtF(t) =˙ 1 T

Z T 0

dt

hΨ(t)|h

H(t)ˆ −i_∂t^∂i

|Ψ(t)i hΨ(t)|Ψ(t)i It satisfies astationary principlesimilar to the time-independent one:

δ_Ψ(t)Q= 0⇐⇒Ψ(t) is a solution to (1)

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The quasi-energy can be expanded with respect to the perturbation strengthsx⁺andx⁻: Q=Q⁽⁰⁾+Q⁽¹⁰⁾x⁺+Q⁽⁰¹⁾x⁻+1

2Q⁽²⁰⁾x⁺²+Q⁽¹¹⁾x⁺x⁻+1

2Q⁽⁰²⁾x⁻²+· · · whereQ⁽⁰⁾=E⁽⁰⁾is the unperturbed energy and the quasi-energy derivatives

Q⁽¹⁰⁾= ∂Q

∂x⁺

x=0

; Q⁽⁰¹⁾= ∂Q

∂x⁻

x=0

; Q⁽²⁰⁾= ∂²Q

∂x⁺²

x=0

; Q⁽¹¹⁾= 2 ∂²Q

∂x⁺∂x⁻

x=0

; etc...

areω-dependent dynamic molecular properties

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2Q⁽²⁰⁾x⁺²+Q⁽¹¹⁾x⁺x⁻+1

Q⁽¹⁰⁾= ∂Q

∂x⁺

x=0

; Q⁽⁰¹⁾= ∂Q

∂x⁻

x=0

; Q⁽²⁰⁾= ∂²Q

∂x⁺²

x=0

; Q⁽¹¹⁾= 2 ∂²Q

∂x⁺∂x⁻

x=0

; etc...

We will consider the specific case of theelectric dipole interaction perturbation H(t) = ˆˆ H₀−E⁺·µˆe^−iωt−E⁻·µˆe^+iωt

where ˆµis the dipole moment operator andE⁺=E⁻ is the amplitude of the oscillating electric field. Thedynamic electric dipole polarizabilityis defined as:

αi,j(ω) =− ∂²Q

∂E_i⁻∂E_j⁺

E=0

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2Q⁽²⁰⁾x⁺²+Q⁽¹¹⁾x⁺x⁻+1

Q⁽¹⁰⁾= ∂Q

∂x⁺

x=0

; Q⁽⁰¹⁾= ∂Q

∂x⁻

x=0

; Q⁽²⁰⁾= ∂²Q

∂x⁺²

x=0

; Q⁽¹¹⁾= 2 ∂²Q

∂x⁺∂x⁻

x=0

; etc...

We will consider the specific case of theelectric dipole interaction perturbation H(t) = ˆˆ H₀−E⁺·µˆe^−iωt−E⁻·µˆe^+iωt

where ˆµis the dipole moment operator andE⁺=E⁻ is the amplitude of the oscillating electric field. Thedynamic electric dipole polarizabilityis defined as:

αi,j(ω) =− ∂²Q

∂E_i⁻∂E_j⁺

E=0

Using time-dependent perturbation theory, we obtain the exact expression:

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In a time-dependent electronic-structure method, the quasi-energyQ(E⁺,E⁻) is obtained by plugging optimal parametersp^o(E⁺,E⁻) in a quasi-energy function Q(E⁺,E⁻;p)

Q(E⁺,E⁻) = Q(E⁺,E⁻;p^o(E⁺,E⁻))

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Q(E⁺,E⁻) = Q(E⁺,E⁻;p^o(E⁺,E⁻))

In contrast to the static case, the parameters now depend on time and take complex values. They can be decomposed in Fourier modes

p=p⁺e^−iωt+p⁻^∗e^+iωt

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Q(E⁺,E⁻) = Q(E⁺,E⁻;p^o(E⁺,E⁻))

In contrast to the static case, the parameters now depend on time and take complex values. They can be decomposed in Fourier modes

p=p⁺e^−iωt+p⁻^∗e^+iωt

We will consider only the case where all the parameters arevariational, i.e. fulfilling the stationary conditions

∂Q

∗

= 0 and ∂Q

∗

= 0

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Thanks to the stationarity of the parameters, only the explicit dependence onE⁺,E⁻ in Q contributes for first-order properties:

∂Q

∂E_i⁻ = ∂Q

∂E_i⁻ + ∂Q

∂p⁺

∂E_i⁻+ ∂Q

∂p⁻

∂E_i⁻ + c.c.

| {z }

= 0

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∂Q

∂E_i⁻ = ∂Q

∂E_i⁻ + ∂Q

∂p⁺

∂E_i⁻+ ∂Q

∂p⁻

∂E_i⁻ + c.c.

| {z }

= 0

For second-order properties, the implicit dependence onE⁺,E⁻via the parameters contributes. In particular, for thedynamic polarizability, we get:

αi,j(ω) =− ∂²Q

∂E_i⁻∂E_j⁺ =− ∂²Q

∂p⁺∂E_i⁻

∂p⁺

∂E_j⁺ − ∂²Q

∂p⁻∂E_i⁻

∂p⁻

∂E_j⁺

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∂Q

∂E_i⁻ = ∂Q

∂E_i⁻ + ∂Q

∂p⁺

∂E_i⁻+ ∂Q

∂p⁻

∂E_i⁻ + c.c.

| {z }

= 0

αi,j(ω) =− ∂²Q

∂E_i⁻∂E_j⁺ =− ∂²Q

∂p⁺∂E_i⁻

∂p⁺

∂E_j⁺ − ∂²Q

∂p⁻∂E_i⁻

∂p⁻

∂E_j⁺

The linear-response vector (∂p⁺/∂E_j⁺, ∂p⁻/∂E_j⁺) is found from thelinear-response equations: 



∂²Q

∂p^+∗∂p⁺

∂²Q

∂p^+∗∂p⁻

∂²Q

∂p^{− ∗}∂p⁺

∂²Q

∂p^{− ∗}∂p⁻









∂p⁺

∂E_j⁺

∂p⁻

∂E_j⁺



=−





∂²Q

∂E_j⁺∂p^+∗

∂²Q

∂E_j⁺∂p^{− ∗}





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∂Q

∂E_i⁻ = ∂Q

∂E_i⁻ + ∂Q

∂p⁺

∂E_i⁻+ ∂Q

∂p⁻

∂E_i⁻ + c.c.

| {z }

= 0

αi,j(ω) =− ∂²Q

∂E_i⁻∂E_j⁺ =− ∂²Q

∂p⁺∂E_i⁻

∂p⁺

∂E_j⁺ − ∂²Q

∂p⁻∂E_i⁻

∂p⁻

∂E_j⁺

The linear-response vector (∂p⁺/∂E_j⁺, ∂p⁻/∂E_j⁺) is found from thelinear-response equations: 



∂²Q

∂p^+∗∂p⁺

∂²Q

∂p^+∗∂p⁻

∂²Q

∂p^{− ∗}∂p⁺

∂²Q

∂p^{− ∗}∂p⁻









∂p⁺

∂E_j⁺

∂p⁻

∂E_j⁺



=−





∂²Q

∂E_j⁺∂p^+∗

∂²Q

∂E_j⁺∂p^{− ∗}





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TheTDHF quasi-energy functionis Q_TDHF(E⁺,E⁻;κ) = 1

T Z T

0

dt hΦ(κ)|H(t)ˆ −i_∂t^∂|Φ(κ)i hΦ(κ)|Φ(κ)i

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T Z T

0

After calculating all the derivatives ofQTDHFwith respect to the orbital rotation parametersκar, we get theTDHF polarizability(for real-valued orbitals):

α^TDHF_i,j (ω) = µ_i

µ_i T

A B

B A

−ω

1 0 0 −1

−1 µ_j µ_j

whereAandBare the same matrices introduced in static response theory (i.e., CPHF)

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T Z T

0

µ_i T

A B

B A

−ω

1 0 0 −1

−1 µ_j µ_j

whereAandBare the same matrices introduced in static response theory (i.e., CPHF) TheTDHF excitation energiesωncorresponds to the poles ofα^TDHFi,j (ω) inω(i.e., the values ofωwhereα^TDHF_i,j (ω) diverges). They are found by solving thepseudo-Hermitian eigenvalue equation:

A B

B A

Xn

Yn

=ωn

1 0 0 −1

Xn

Yn

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T Z T

0

µ_i T

A B

B A

−ω

1 0 0 −1

−1 µ_j µ_j

whereAandBare the same matrices introduced in static response theory (i.e., CPHF) TheTDHF excitation energiesωncorresponds to the poles ofα^TDHFi,j (ω) inω(i.e., the values ofωwhereα^TDHF_i,j (ω) diverges). They are found by solving thepseudo-Hermitian eigenvalue equation:

A B

B A

Xn

Yn

=ωn

1 0 0 −1

Xn

Yn

Theoscillator strengthscan be calculated from the eigenvectors (Xn,Yn) fn=2

3ωn

X

i

h

(Xn+Yn)^Tµ_i i2

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TheTDDFT quasi-energy functionis Q_TDDFT(E⁺,E⁻;κ) = 1

T ZT

0

dt hΦ(κ)|ˆh(t)−i_∂t^∂|Φ(κ)i

hΦ(κ)|Φ(κ)i +EH[ρ_Φ(κ)]

!

+ Q_xc[ρ_Φ(κ)]

where ˆh(t) is the one-electron Hamiltonian and Q_xc[ρ] is theexchange-correlation quasi-energy functionaldepending on the time-dependent densityρ

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T ZT

0

!

+ Q_xc[ρ_Φ(κ)]

In practice, we always use theadiabatic approximation Q_xc[ρ]≈ 1

T Z T

0

dt Exc[ρ]

whereExc[ρ] is the ground-state exchange-correlation functional

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T ZT

0

!

+ Q_xc[ρ_Φ(κ)]

In practice, we always use theadiabatic approximation Q_xc[ρ]≈ 1

T Z T

0

dt Exc[ρ]

whereExc[ρ] is the ground-state exchange-correlation functional

We then obtain the same equations as in TDHF with the same matricesAandBas in CPKS

A = (ε −ε )δ δ +hrb|asi+hrb|f |asi and B =hrs|abi+hrs|f |abi

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TheTamm-Dancoff approximation (TDA)corresponds to neglecting the matrixB.

The TDHF or TDDFT linear-response equations then simplifies to A Xn=ωnXn

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In TDHF, the matrixAcorresponds to the matrix of the Hamiltonian (shifted by the HF energy) in the basis of the single-excited determinants: Aar,bs=hΦ^ra|Hˆ0−EHF|Φ^s_bi. We thus recover aconfiguration-interaction singles (CIS)calculation.

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In TDHF, the matrixAcorresponds to the matrix of the Hamiltonian (shifted by the HF energy) in the basis of the single-excited determinants: Aar,bs=hΦ^ra|Hˆ0−EHF|Φ^s_bi. We thus recover aconfiguration-interaction singles (CIS)calculation.

In the TDA, we obtainexcited-state wave functionsas

|Ψni= Xocc

a

Xvir r

Xn,ar|Φ^rai

whereXn,ar is the coefficient for the single orbital excitationa→r. Usually, one orbital excitation dominates over all the other orbital excitations, and this is used to “assign” the excited-state state to this orbital excitation.

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Valence and Rydberg excitation energies of the CO molecule (Sadlej+ basis set)

State Transition TDHF TDHF-TDA TDLDA TDLDA-TDA EOM-CCSD Expt Valence excitation energies (eV)

3Π 5σ→2π^∗ 5.28 5.85 5.95 6.04 6.45 6.32

3Σ⁺ 1π→2π^∗ 6.33 7.79 8.38 8.54 8.42 8.51

1Π 5σ→2π^∗ 8.80 9.08 8.18 8.42 8.76 8.51

3∆ 1π→2π^∗ 7.87 8.74 9.16 9.20 9.39 9.36

3Σ⁻ 1π→2π^∗ 9.37 9.73 9.84 9.84 9.97 9.88

1Σ⁻ 1π→2π^∗ 9.37 9.73 9.84 9.84 10.19 9.88

1∆ 1π→2π^∗ 9.96 10.15 10.31 10.33 10.31 10.23

3Π 4σ→2π^∗ 13.05 13.31 11.40 11.43 12.49

Rydberg excitation energies (eV)

3Σ⁺ 5σ→6σ 11.07 11.18 9.55 9.56 10.60 10.40

1Σ⁺ 5σ→6σ 12.23 12.27 9.93 9.95 11.15 10.78

3Σ⁺ 5σ→7σ 12.40 12.42 10.26 10.26 11.42 11.30

1Σ⁺ 5σ→7σ 12.78 12.79 10.47 10.50 11.64 11.40

3Π 5σ→3π 12.52 12.60 10.39 10.39 11.66 11.55

1Π 5σ→3π 12.87 12.88 10.48 10.50 11.84 11.53

Ionization threshold:−ǫ_HOMO(eV)

15.11 15.11 9.12 9.12 15.58

Mean absolute deviations of excitation energies with respect to EOM-CCSD (eV)

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There are on-going efforts for improving the approximations used in linear-response TDDFT:

Hybridapproximations (mixing HF exchange and DFT exchange) andrange-separated hybridapproximations (mixing long-range HF exchange and short-range DFT exchange) improve the description of nonlocal excitations(e.g., Rydberg and charge transfer)

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Hybridapproximations (mixing HF exchange and DFT exchange) andrange-separated hybridapproximations (mixing long-range HF exchange and short-range DFT exchange) improve the description of nonlocal excitations(e.g., Rydberg and charge transfer) A number of double-hybridapproximations (mixing PT2 correlation and DFT

correlation) for TDDFT have been proposed

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The TDDFT-like approach known as theBethe-Salpeter equation(BSE), originally coming from condensed-matter physics, is increasing applied to molecules. It is similar to TDHF but with a screened HF exchange kernel accounting for correlation

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The TDDFT-like approach known as theBethe-Salpeter equation(BSE), originally coming from condensed-matter physics, is increasing applied to molecules. It is similar to TDHF but with a screened HF exchange kernel accounting for correlation

Situations withstatic/strong correlation(bond dissociation, transition metals, conical intersection, ...) are usually a problem in TDDFT with the usual approximations. A number of approaches combiningmultideterminant wave function methods(e.g.,