Gauge-origin invariant calculations of excitation energies ... · PDF fileGauge-origin...

Gauge-origin invariant calculations of excitationenergies for molecules subject to strong magnetic

fields

Erik Tellgren

Center for Theoretical and Computational Chemistry, Department of Chemistry,University of Oslo, Norway

Workshop on Quantum Chemistry in Strong Magnetic Fields,September 13–14, 2010

Gauge-origin invariant calculations of excitationenergies (and geometrical gradients) for molecules

subject to strong magnetic fields

Erik Tellgren

Center for Theoretical and Computational Chemistry, Department of Chemistry,University of Oslo, Norway

Workshop on Quantum Chemistry in Strong Magnetic Fields,September 13–14, 2010

Gauge orig.inv., excitations, gradients. . . 2/35

Outline

1 The London programMagnetic Periodic Boundary Conditions, hybrid basis setsReminder about gauge (in)variance, London orbitals

2 Finite magnetic fields in quantum chemistry: How strong?

3 Hartree–Fock ground state results

4 Random phase approximation for excitationsFormalism, technicalitiesSome preliminary results

5 Differentiated integrals and related issuesGeometry optimization

6 Summary

Erik Tellgren Gauge orig.inv., excitations, gradients. . . 2/35


The London program

Magnetic Periodic Boundary Conditions, hybrid basis sets

From MPBC to mixed basis sets & London orbitals

(Collaboration with A. Soncini)

A Gaussian-type orbital approach to periodic systems in(finite) uniform magnetic fields. . .

. . . leads to integrals over mixed plane-wave/GTO functions:

ω(r) = (x − Ax)l(y − Ay )m(z − Az)ne−a(r−A)2e−iq·r (1)

From a method-development P.O.V., no difference betweenmatrix element evaluation for:

periodic systems, finite magnetic fields, GTOsperiodic systems, finite magnetic fields, London orbitalsmolecular systems, finite magnetic fields, London orbitalsmolecular systems, mixed plane-wave/GTO basis sets



The London program











The London program











The London program



3rd alt.: molecular systems, finite magnetic fields, Londonorbitals

Interesting in itself: Excepting a few studies on very smallsystems (H2, Be atom,. . . ), it has not been done before

Proof of concept for integrals arising in other cases

All integrals now become complex-valued. In particular,Coulomb integrals involve the Boys function of acomplex-valued argument1

Fn(z) =

∫ 1

0tne−zt2

dt (2)

Hard to reuse standard packages (e.g. Dalton), due theirreliance on real-valued quantities

1T. N. Rescigno et al. Phys Rev A 11:825 (1975) and N. S. Ostlund ChemPhys Lett 34:419 (1975) for scattering calculations



The London program







Fn(z) =

∫ 1

0tne−zt2

dt (2)





The London program







Fn(z) =

∫ 1

0tne−zt2

dt (2)





The London program







Fn(z) =

∫ 1

0tne−zt2

dt (2)





The London program


London orbitals: implementation issues

Brief (and incomplete) review of integral evaluation:

Analytical expressions for Coulomb integrals (scattering):N. Ostlund, Chem Phys Lett 34:419 (1975)

T. Rescigno et al. Phys Rev A 11:825 (1975)

Integration schemes suitable for implementation (planewave/Gaussian hybrid functions):

Rys quadrature: P. Carsky & M. Polasek, J Comp Phys 143:266 (1998)

McMurchie–Davidson scheme (no implementation): M. Tachikawa &

M. Shiga, Phys Rev E 64:056706 (2001)

Integration schemes suitable for implementation (Londonorbitals considered):

Obara–Saika scheme, application to H2: S. Kiribayashi et al. IJQC 75:637

(1999)

ACE scheme (no implementation): K. Ishida, J Chem Phys 118:4819 (2003)

McMurchie–Davidson scheme: E. Tellgren, A. Soncini & T. Helgaker, J Chem

Phys 129:154114 (2008)



The London program

Reminder about gauge (in)variance, London orbitals

London orbitals: A vs. B

A gauge transformation using an arbitrary f (r, t),

V ′ = V − ∂f (r, t)

∂t(3)

A′ = A +∇f (r, t) (4)

ψ′ = e if (r,t)ψ (5)

only affects the non-physical degrees of freedom.

A finite basis set does not approximate all gauge transformedwave functions equally well.

How well a wave function ψ′ = e if (r)ψ can be approximateddepends on the gauge function f (r).

It is hopeless to achieve gauge invariance using Gaussian-typebasis sets of reasonable size.



The London program




V ′ = V − ∂f (r, t)

∂t(3)

A′ = A +∇f (r, t) (4)

ψ′ = e if (r,t)ψ (5)







The London program




V ′ = V − ∂f (r, t)

∂t(3)

A′ = A +∇f (r, t) (4)

ψ′ = e if (r,t)ψ (5)







The London program


London orbitals: gauge invariance and finite basis sets

Example: H2 molecule, on the x-axis, in the field B = 110 z.

A = 120 z× r −→ A′ = A +∇(10x2)

ψ = RHF/aug-cc-pVQZ −→ ψ′ = e i ·10x2ψ

(6)

−1.5 −1 −0.5 0 0.5 1 1.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Space coordinate, x (along the bond)

Wav

e fu

nctio

n, ψ

Re(ψ)Im(ψ)

|ψ|2

−1.5 −1 −0.5 0 0.5 1 1.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5


Gau

ge tr

ansf

orm

ed w

ave

func

tion,

ψ′

Re(ψ′)Im(ψ′)|ψ′|2



The London program


London orbitals: gauge invariance vs. gauge-origininvariance

Gauge invariance is not practically feasible with Gaussian basissets.

Restrict attention to uniform fields, fix some of the gaugefreedom by choosing ∇ · A = 0, and in addition take A to becylindrically symmetric.

A magnetic field B0 is then represented byA(r) = 1

2B0 × (r − G).

The gauge origin G contains the remaining gauge degrees offreedom.

More modest goal: make finite basis set calculationsindependent of the gauge origin G.



The London program


London orbitals: definition of

Consider an atomic orbital χao(r) centered at C.

Define the London orbital

χlo(r) = e−iA(C)·rχao(r) (7)

as the AO times a gauge factor.

When the AOs are Gaussian-type orbitals, the London orbitalstake the form

χ(r) = x lymzne−γ(r−C)2−iA(C)·r. (8)

London orbitals make all quantities gauge-origin independent.



The London program


London orbitals: illustration

Example: H2 molecule, on the x-axis, in the field B = 110 z.

A = 120 z× r −→ A′ = A +∇(−A(G) · r)

ψ = RHF/aug-cc-pVQZ −→ ψ′ = e−iA(G)·rψ(9)

Gauge-origin moved from 0 to G = 100y.

−1.5 −1 −0.5 0 0.5 1 1.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5


Wav

e fu

nctio

n, ψ

Re(ψ)Im(ψ)

|ψ|2

−1.5 −1 −0.5 0 0.5 1 1.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5


Gau

ge tr

ansf

orm

ed w

ave

func

tion,

ψ′

Re(ψ′)Im(ψ′)|ψ′|2



The London program


The London program

The London program is an ab initio program for finite fieldcalculations using London orbitals2:

Hartree-Fock wave functions [RHF, UHF, GHF].

DFT version?

Quite general integral evaluation

Excitation energies using RPA

Geometrical gradients and geometry optimization functionality

Automatic generation of current density grid files

Not yet highly optimized for speed. C20 is a “large” system.

2E. Tellgren, A. Soncini, T. Helgaker, J Chem Phys, 129:154114 (2008)Erik Tellgren Gauge orig.inv., excitations, gradients. . . 11/35


Finite magnetic fields in quantum chemistry: How strong?

Magnetic field strength in context

1 T is a large unit, 1 au = 2.35× 105 T

National High Magnetic Field Lab (Florida): 45 T (sustained),88.9 T (2 sec.), 1000 T (∼ 1 ms)3

For finite field calculations on diatomic molecules, ∼ 0.1 au isa natural field. . .

. . . shows non-linear response clearly

Larger molecules can be probed with smaller fields

3http://www.magnet.fsu.edu/education/tutorials/magnetacademy/magnets/page7.htmlErik Tellgren Gauge orig.inv., excitations, gradients. . . 12/35





































Earth magnetic field (30-50 microT)

fridge magnet

1 mT4.26e-9 au

field in sunspot

loud-speaker magnet medical MRI

1 T4.26e-6 au

500 MHz NMR spectro. (11.7 T)frog levitation exper. (16 T) strongest static field (45 T)

non-destructive pulse (88.9 T)

expl. pulsed fields (2.8 kT)

1 kT0.00426 au

neutron star, 1-100 MT

1 MT4.26 au

magnetar, up to 100 GTBrel = 4.41 GT,

cyclotron energy = mc^2

1 GT4260 au


Hartree–Fock ground state results

Hypermagnetizabilities (benzene)

compute points on E (B) curve, least-squares fit a polynomial−→ estimate of Taylor coeff.

Benzene (C6H6) finite field results:

basis set χxx χyy χzz Xxxxx Xyyyy Xzzzz

London orbitalsSTO-3G −8.11 −8.11 −22.97 −211.17 −211.17 −52.086-31G −8.24 −8.24 −23.14 −218.79 −218.79 −64.32cc-pVDZ −8.08 −8.08 −22.27 −235.96 −235.96 −120.11aug-cc-pVDZ −7.99 −7.99 −22.41 −316.46 −316.46 −152.67

common-origin: centre of massSTO-3G −35.75 −35.75 −48.11 45.21 45.21 26.916-31G −31.56 −31.56 −39.42 28.88 28.88 −151.57cc-pVDZ −15.38 −15.38 −26.93 8.90 8.90 −241.48aug-cc-pVDZ −9.91 −9.91 −25.22 −412.51 −412.51 −158.50

common-origin: hydrogen nucleusSTO-3G −35.75 −176.25 −116.69 45.19 1477 −53406-31G −31.56 −144.81 −88.03 28.88 1588 −5866cc-pVDZ −15.38 −48.00 −41.56 9.11 2935 −3355aug-cc-pVDZ −9.91 −20.94 −33.86 −412.55 −3321 −1097

(N.B.: decontracted basis sets)Erik Tellgren Gauge orig.inv., excitations, gradients. . . 14/35



Non-perturbative phenomena

Benzene illustrates the typical caseSome paratropic (C4H4) and paramagnetic (BH, C20 ring)systems have slowly converging/diverging perturbationexpansions

−0.1 −0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

∆ E

(B)

C6H

6, aug−cc−pVDZ

−0.1 −0.05 0 0.05 0.10

5

10

x 10−3

∆ E

(B)

C4H

4, aug−cc−pVDZ

0 0.1 0.2 0.3 0.40

0.01

0.02

0.03

0.04

0.05

0.06

∆ E

(B)

BH, aug−cc−pVTZ

0 0.01 0.02 0.03 0.04 0.05

−0.03

−0.02

−0.01

0

∆ E

(B)

C20

, cc−pVDZ




Non-perturbative phenomena: perturbative divergence

Polynomials fitted to the BH energy curve (aug-cc-pVDZ):

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−25.165

−25.16

−25.155

−25.15

−25.145

−25.14

−25.135

−25.13 data pointsdegree 6degree 10degree 14

Perturbative response calculations need to go to order > 14.

A two-state model (2× 2 Hamiltonian) accounts for thisbehavior (as well as the diamagnetic case)4

4E. Tellgren, T. Helgaker, A. Soncini PCCP 11:5489 (2009)Erik Tellgren Gauge orig.inv., excitations, gradients. . . 16/35





0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−25.165

−25.16

−25.155

−25.15

−25.145

−25.14

−25.135









0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−25.165

−25.16

−25.155

−25.15

−25.145

−25.14

−25.135






Random phase approximation for excitations

Formalism, technicalities

RPA formulation & implementation for London

Desiderata:linear basis transformations shouldn’t change the form of the working equations5 (“covariance”)

allow any basis (e.g. AOs), avoid MO basis in practice

don’t split complex quantities into real- and imaginary parts

unified formulation & implementation for RHF, UHF and GHF

Sample of recent non-MO formulations of RPA (TD-HF):H. Larsen et al. JCP 113:8908 (2000), S. Coriani et al. JCP 126:154108 (2007)

M. Lucero et al. JCP 129:064114 (2008), T. Kjærgaard et al. JCP 129:054106 (2008)

Rederiving RPA with the desiderata in mind, lead to aformulation essentially identical to that of Kjærgaard et al.Minor differences: complex quantities, notation reflects

“covariance”, indices may refer to either space orbitals, spin

orbitals, 2-comp. orbitals.

Davidson’s method iteratively solves the eigenvalue problem,exploiting the paired structure of excitation operators

5Head-Gordon et al. J Chem Phys 108:616 (1998)Erik Tellgren Gauge orig.inv., excitations, gradients. . . 17/35

















































Different paths to RPA

Hartree-Fock state|0>

TD-HF<<A; B>>

time-dep. pert.

RPA exc.<0| [[H,Epq],X] |0> = w <0| [X,Epq] |0>

resp.fun. poles

CID state|c> = a|0>+...

approx. exc.[H,X] |c> = w X |c>

1-el. exc.op.X = Xpq Epq

approx. exc.<c| [[H,Epq],X] |c> = w <c| [X,Epq] |c>

projection

1st order in (1-a)






Very tedious but straightforward to work out

W[2]ζη,θκ = 〈0|[[H, Eζη], Eθκ]|0〉 and S

[2]ζη,θκ = 〈0|[Eθκ, Eζη]|0〉

explicitly in AO basis

But only their effects on a trial vector are needed

Hessian and metric transformations

W[2]ζη,θκX θκ = −[[P, X ], F (P)]ηζ − [P, G ([P,X ])]ηζ (10)

S[2]ζη,θκX θκ = [P, X ]ηζ (11)




Some preliminary results

RPA spectrum of boronmonohydride (BH)

Singlet and triplet energies vs. field for BH [RHF/aug-cc-pVDZ]

0.00 0.15 0.30 0.45

−25.1

−25.0

−24.9

−24.8

−24.7

0.00 0.15 0.30 0.45

−25.1

−25.0

−24.9

−24.8

−24.7

The field is perpendicular to the bond axis. Instabilities atB⊥ ≈ 0.1, 0.16 au.







0.00 0.15 0.30 0.45

−25.1

−25.0

−24.9

−24.8

−24.7

0.00 0.15 0.30 0.45

−25.1

−25.0

−24.9

−24.8

−24.7

The field is perpendicular to the bond axis. Instabilities atB⊥ ≈ 0.1, 0.16 au.







0.00 0.15 0.30 0.45

−25.1

−25.0

−24.9

−24.8

−24.7

0.00 0.15 0.30 0.45

−25.1

−25.0

−24.9

−24.8

−24.7

The field is parallel to the bond axis. Lowest triplets invisible atB‖ = 0 au.







0.00 0.15 0.30 0.45

−25.1

−25.0

−24.9

−24.8

−24.7

0.00 0.15 0.30 0.45

−25.1

−25.0

−24.9

−24.8

−24.7

The field is parallel to the bond axis. Lowest triplets invisible atB‖ = 0 au.






Singlet dip. oscillator strengths vs. field for BH

0.00 0.15 0.30 0.45

−25.1

−25.0

−24.9

−24.8

−24.7

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.450.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Exc 1Exc 2Exc 3

Field is perpendicular to the bond axis. Osc.str. for lowest 3 exc.shown.





Lowest triplet in H2: Basis set convergence

Energy (left) and dipole trans.mom. (right) vs. perpendicular field.

0.00 0.15 0.30 0.450.26

0.28

0.30

0.32

0.34

0.36

0.38

0.40

0.42STO-3G6-311++G**aug-cc-pVDZaug-cc-pVTZ

0.00 0.15 0.30 0.451.30

1.35

1.40

1.45

1.50

1.55

1.60

1.65

1.70

STO-3G6-311++G**aug-cc-pVDZaug-cc-pVTZ

Solid lines: London orbitalsDashed lines: Gaussian orbitals (g.o. on H)





Methyl radical CH3 (UHF/aug-cc-pVDZ)

Left: Energy spectrum vs. field [dashed lines dip. forbidden]Right: Dip. osc. strength vs. field

0.00 0.15 0.30 0.45−39.65

−39.60

−39.55

−39.50

−39.45

−39.40

−39.35

−39.30

−39.25

0.00 0.15 0.30 0.450.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

velocity gauge and length gaugeNo spin-flipped states in the above spectrumA magnetic field modulates dipole transitions! (RPA approx.)







0.00 0.15 0.30 0.45−39.65

−39.60

−39.55

−39.50

−39.45

−39.40

−39.35

−39.30

−39.25

0.00 0.15 0.30 0.450.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07








0.00 0.15 0.30 0.45−39.65

−39.60

−39.55

−39.50

−39.45

−39.40

−39.35

−39.30

−39.25

0.00 0.15 0.30 0.450.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07








0.00 0.15 0.30 0.45−39.65

−39.60

−39.55

−39.50

−39.45

−39.40

−39.35

−39.30

−39.25

0.00 0.15 0.30 0.450.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07




Differentiated integrals and related issues

Differentiated GTO integrals

Hermite scheme for differentiated GTO integrals6

Expands spherical GTOs in Hermite GTOs rather thanCartesian GTOs

Exploits special properties of Hermite GTOs

Simple unified implementation of arbitrary order derivatives

(Some other benefits as well, some speed-up . . . )

Not obvious how to generalize to London orbitals

Is there a (analogous or disanalogous) way to achieve the sameunification & simplicity for finite-field London orbitals?

6S. Reine, E. Tellgren, T. Helgaker PCCP 9:4771 (2007)Erik Tellgren Gauge orig.inv., excitations, gradients. . . 25/35















































One approach to differentiated London orbitals

Think of a basis set as a pair B = (M,P)

P is a set of primitive GIAOs

M is a linear basis mapping that. . .

. . . defines linear combinations of prim. GIAOs

. . . is typically highly block diagonal in a matrix repr.

Some examples of such mappings:

Cartesian-to-spherical transformation: maps prim. GIAOs tolin.comb. within the same (sub)shell

Contraction: maps prim. GIAOs to lin.comb. with sameang.mom., different exponents

If block-diagonality isn’t required, also:

OrthogonalizationSymmetry adaptation

























































































Some other examples:

Magnetic-field gradient: M1[χi ] = ∂χi/∂B

Geometrical gradient: M2[χi ] = ∂χi/∂R

Geometrical Hessian: M3[χi ] = M2[M2[χi ]]

Kinetic-/magnetic-balance: M4[χiωspin] = σ · πχiω

spin

Compositions of the above, e.g. Mderiv ◦Mcontr

Given B1 = (M1,P1) and B2 = (M2,P2):

Compute matrix elements between P1 and P2

Transform to mat.elem. between B1 and B2 using M1 and M2

(Can be done block-by-block if M1 and/or M2 is block diag.)










spin


Given B1 = (M1,P1) and B2 = (M2,P2):













spin


Given B1 = (M1,P1) and B2 = (M2,P2):













spin


Given B1 = (M1,P1) and B2 = (M2,P2):













spin


Given B1 = (M1,P1) and B2 = (M2,P2):













spin


Given B1 = (M1,P1) and B2 = (M2,P2):













spin


Given B1 = (M1,P1) and B2 = (M2,P2):













spin


Given B1 = (M1,P1) and B2 = (M2,P2):








When higher-order derivatives/several basis sets are involved,another operation is useful:

Given B1 = (M1,P1) and B2 = (M2,P2), reexpress asB′1 = (M ′

1,P1 ∪ P2) and B′2 = (M ′2,P1 ∪ P2)

Mi [Pi ] = M ′i [P1 ∪ P2] =⇒ Bi ,B′i define the same basis

functions

Enables reuse of intermediate integrals (e.g. gradients for freewhen computing Hessians)




Geometry optimization

Current status

Geometrical gradients have been implemented in Londonusing the above scheme

(When the density matrix response is available, it should alsobe easy to implement Hessians.)

BFGS geometry optimization implemented

works OK, but struggles to rotate molecules into optimalalignment with the external fieldneed to incorporate some extra coordinate that respondsdirectly to the total torquefor now, it’s best to provide an initial guess with the desiredalignment





Current status









Current status









Current status









Current status









Example: H2O geometry in strong fields

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1693.95

94

94.05

94.1

94.15

94.2

94.25

94.3

B [au]

d(O

−H

) [p

m]

H2O, HF/6−31G**

oopin−plane, ⊥ C2−axisin−plane, || C2−axis

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16105.4

105.6

105.8

106

106.2

106.4

106.6

106.8

107

B [au]

H−

O−

H a

ngle

[deg

rees

]

H2O, HF/6−31G**

oopin−plane, ⊥ C2−axisin−plane, || C2−axis

bonds contract

for B parallel to the C2-axis, the bond angle decreases(reducing the diamagnetic term)

otherwise the bond angle increases





Example: NH3 geometry in strong fields

Field along symmetry axis:

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1699.65

99.7

99.75

99.8

99.85

99.9

99.95

100

100.05

100.1

100.15

B [au]

d(N

−H

) [p

m]

NH3, HF/6−31G**

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16107.5

107.6

107.7

107.8

107.9

108

108.1

108.2

108.3

B [au]

H−

N−

H a

ngle

[deg

rees

]

NH3, HF/6−31G**

bonds contract

bond angle increases −→ more planar geometry





Example: CH4 geometry in strong fields

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16107.95

108

108.05

108.1

108.15

108.2

108.25

108.3

108.35

108.4

B [au]

d(C

−H

) [p

m]

CH4, HF/6−31G**

|| C2−axis|| C3−axis, C−H|| C3−axis

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16107.5

108

108.5

109

109.5

110

110.5

B [au]

H−

C−

H a

ngle

[deg

rees

]

CH4, HF/6−31G**

|| C2−axis|| C2−axis|| C3−axis|| C3−axis

C2 case: reduces extent ⊥ B & the diamagnetic term

C3 case: angle between bond parallel to the field & otherbonds decreases −→ C & the 3 other H become more planar

C3 case: hint of non-linearity in crossing bond length curves


Example: C6H6 geometry in strong fields

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16106.8

106.9

107

107.1

107.2

107.3

107.4

107.5

107.6

B [au]

d(C

−H

) [p

m]

C6H

6, HF/6−31G*

C−H || BC−H other

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16138.3

138.35

138.4

138.45

138.5

138.55

138.6

138.65

138.7

138.75

B [au]

d(C

−C

) [p

m]

C6H

6, HF/6−31G*

C−C other (c)C−C || B (d)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16117

117.5

118

118.5

119

119.5

120

120.5

B [au]

bond

ang

le [d

egre

es]

C6H

6, HF/6−31G*

C−C−C (a)C−C−H (b)

Fits with Caputo & Lazzeretti IJQC (online 31 aug 2010)


Summary

Summary

London opens up for several applications, from analternative method to compute static response properties toinvestigation of intrinsically non-perturbative phenomena

Functionality for RPA excitations recently added

AO-/arbitrary basis formulationunified handling of RHF/UHF/GHF

Functionality related to gradients recently added

part of quite simple & general scheme for derivativesBFGS geometry optimizeropens up for Hessians, 4-component integrals

Future goals include

Correlation: CI, etc.Study of CDFT functionals using Adiabatic Connectionmethods (with A. Teale)



Summary

Summary










Summary

Summary










Summary

Summary










Acknowledgments

Thanks to. . .

Trygve Helgaker

Alessandro Soncini

Kai Kaarvann Lange (master student working on the RPAimplementation)

Pawe l Sa lek, Simen Reine

The audience


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