Derivatives and Properties - Universitetet i oslofolk.uio.no/helgaker/talks/MWM16_DerivativesProperties.pdf · Trygve Helgaker (CTCC, University of Oslo) Derivatives and Properties

Derivatives and Properties

Trygve Helgaker

Centre for Theoretical and Computational Chemistry (CTCC),Department of Chemistry, University of Oslo, Norway

Summer School: Modern Wavefunction Methods in Electronic Structure TheoryWissenschaftspark Gelsenkirchen, Gelsenkirchen, Germany

October 3–8, 2016

Trygve Helgaker (CTCC, University of Oslo) Derivatives and Properties MWM Summer School 2016 1 / 87

Time-Independent Molecular Properties

I When a molecular system is perturbed, its total energy changes

E(µ) = E(0) + E(1)µ+ 12E(2)µ2 + · · ·

I The expansion coefficients are characteristic of the molecule and its quantum state

I we refer to these coefficients as molecular properties

I When the perturbation is static, the properties may be calculated by differentiation

E(1) =dEdµ

∣∣∣∣µ=0

E(2) =d2Edµ2

∣∣∣∣µ=0

I such properties are said to be time independent

I We do not here consider time-dependent molecular properties

I if periodic, these can be calculated in similar way using the quasi-energy


Examples of Derivatives

I Responses to geometrical perturbations

I forces and force constantsI spectroscopic constants

I Responses to external electromagnetic fields

I permanent and induced momentsI polarizabilities and magnetizabilitiesI optical activity

I Responses to external magnetic fields and nuclear magnetic moments

I NMR shielding and indirect spin–spin coupling constantsI EPR hyperfine coupling constants and g values

I Responses to nuclear quadrupole moments

I nuclear field gradients, quadrupole coupling constants

I Responses to molecular rotation

I spin–rotation constants and molecular g values


Overview

1 Analytical Derivative Theory

2 Geometrical Properties

3 Electronic Hamiltonian

4 London Orbitals

5 Zeeman and Hyperfine Interactions

6 Molecular Magnetic Properties


Analytical Derivative Theory

Table of Contents

1 Analytical Derivative TheoryEnergy FunctionDerivatives for Variational Wave FunctionsDerivatives for Nonvariational Wave FunctionsLagrangian MethodHartree–Fock Molecular GradientFCI Lagrangian and DerivativesSecond-Quantization Hamiltonian for Derivatives

2 Geometrical PropertiesBond DistancesHarmonic and Anharmonic Constants

3 Electronic HamiltonianParticle in a Conservative Force FieldParticle in a Lorentz Force FieldElectron SpinMolecular Electronic Hamiltonian

4 London OrbitalsGauge-Origin TransformationsLondon Orbitals

5 Zeeman and Hyperfine InteractionsParamagnetic OperatorsHamiltonian with Zeeman and Hyperfine OperatorsDiamagnetic Operators

6 Molecular Magnetic PropertiesFirst-Order Magnetic PropertiesMolecular MagnetizabilitiesNMR Spin HamiltonianNuclear Shielding ConstantsIndirect Nuclear Spin–Spin Coupling Constants


Analytical Derivative Theory

Numerical vs. analytical differentiation

I Numerical differentiation (finite differences and polynomial fitting)

I often simple to implement (at least for real singlet perturbations)I difficulties related to numerical accuracy and computational efficiency

I Analytical differentiation (derivatives calculated from analytical expressions)

I considerable programming effort requiredI greater speed, precision, and convenience

I Implementations of analytical techniques

I first-order properties (dipole moments and gradients)I second-order properties (polarizabilities and Hessians, NMR parameters)


Analytical Derivative Theory Energy Function

Electronic Energy Function

I The electronic energy function contains the Hamiltonian and the wave function:

E(x , λ) = 〈λ |H(x)|λ〉I It depends on two distinct sets of parameters:

x : external (perturbation) parameters (geometry, external field)λ: electronic (wave-function) parameters (MOs, cluster amplitudes)

I The Hamiltonian (here in second quantization)

H(x) =∑pq

hpq(x)Epq + 12

∑pqrs

gpqrs(x)epqrs + hnuc(x)

depends explicitly on the external parameters:

hpq(x) = 〈φp(x) |h(x)|φq(x)〉I The wave function |λ〉 depends implicitly on the external parameters λ(x).



Electronic Energy and its Derivatives

I The electronic energy E(x) is obtained by optimizing the energy function E(x , λ) withrespect to λ for each value of x :

E(x) = E(x , λ∗)

I note: the optimization is not necessarily a variational minimization

I Our task is to calculate derivatives of E(x) with respect to x :

dE(x)

dx=

∂E(x , λ∗)

∂x︸︷︷︸explicit dependence

+∂E(x , λ)

∂λ

∣∣∣∣λ=λ∗

∂λ

∂x

∣∣∣∣λ=λ∗︸︷︷︸

implicit dependence

I the implicit as well as explicit dependence must be accounted for

I The quantity ∂λ/∂x is the wave-function response

I it tells us how the electronic structure changes when the system is perturbed

I To proceed, we need to make a distinction between

I variationally determined wave functionsI nonvariationally determined wave functions



Variational and Nonvariational Wave FunctionsVariational wave functions

I the optimized energy fulfils the stationary (variational) condition:

∂E(x, λ)

∂λ= 0 (for all x)

I the Hartree–Fock energy in an unconstrained exponential parameterization

|HF〉 = exp(−κ)|0〉, κ† = −κ

I the energy of the full CI (FCI) wave function |FCI〉 =∑

i ci |i〉 as an expectation value:

∂EFCI(x, c)

∂c= 0

Nonvariational wave functions

I wave functions whose energy does not fulfil the stationary condition:

∂E(x, λ)

∂λ6= 0

I the Hartree–Fock and Kohn–Sham energies in a constrained LCAO parameterization (orthonormality)

|HF〉 = 1√N!

det|φ1, φ2, . . . φN |, φp(r; x) =∑

µCµpχµ(r; x),

⟨φp |φq

⟩= δpq

I the truncated CI energy with respect to orbital rotations:

∂ECI(x, c, κ)

∂c= 0,

∂ECI(x, c, κ)

∂κ6= 0


Analytical Derivative Theory Derivatives for Variational Wave Functions

Molecular Gradients for Variational Wave Functions

I Applying the chain rule, we obtain for the total derivative of the energy:

dE(x)

dx=∂E(x , λ)

∂x+∂E(x , λ)

∂λ

∂λ

∂x

I the first term accounts for the explicit dependence on xI the last term accounts for the implicit dependence on x

I We now invoke the stationary condition:

∂E(x , λ)

∂λ= 0 (zero electronic gradient for all x)

I The molecular gradient then simplifies to

dE(x)

dx=∂E(x , λ)

∂x

I examples: HF/KS and MCSCF molecular gradients (exponential parameterization)

For variational wave functions, we do not need the response of the wave function∂λ/∂x to calculate the molecular gradient dE/dx .



Hellmann–Feynman Theorem

I Assume that the (stationary) energy is an expectation value:

E(x , λ) = 〈λ |H(x)|λ〉I The gradient is then given by the expectation-value expression:

dE(x)

dx=∂E(x , λ)

∂x=

⟨λ

∣∣∣∣∂H∂x∣∣∣∣λ⟩ ← the Hellmann–Feynman theorem

I Relationship to first-order perturbation theory:

E (1) =⟨

0∣∣∣H(1)

∣∣∣ 0⟩

I The Hellmann–Feynman theorem was originally stated for geometrical distortions:

dEdRK

= −⟨λ

∣∣∣∣∣∑i

ZK riKr3iK

∣∣∣∣∣λ⟩

+∑I 6=K

ZIZKRIK

R3IK

I Classical interpretation: integration over the force operator



Molecular Hessians for Variational Wave Functions

I Differentiating the molecular gradient, we obtain the molecular Hessian:

d2E(x)

dx2=

d

dx

∂E(x , λ)

∂x=

(∂

∂x+∂λ

∂x

∂

∂λ

)∂E(x , λ)

∂x

=∂2E(x , λ)

∂x2+∂2E(x , λ)

∂x∂λ

∂λ

∂x

I we need the first-order response ∂λ/∂x to calculate the HessianI but we do not need the second-order response ∂2λ/∂x2 for stationary energies

I To determine the response, we differentiate the stationary condition:

∂E(x , λ)

∂λ= 0 (all x) =⇒ d

dx

∂E(x , λ)

∂λ= 0

=⇒ ∂2E(x , λ)

∂x∂λ+∂2E(x , λ)

∂λ2

∂λ

∂x= 0

I These are the first-order response equations:

∂2E

∂λ2︸︷︷︸electronicHessian

∂λ

∂x= − ∂2E

∂x∂λ︸︷︷︸right-hand

side



Response Equations

I The molecular Hessian for stationary energies:

d2Edx2

=∂2E

∂x2+

∂2E

∂x∂λ

∂λ

∂x

I The response equations:

electronicHessian → ∂2E

∂λ2

∂λ

∂x= − ∂2E

∂λ∂x←

perturbedelectronic gradient

I the electronic Hessian is a Hermitian matrix, independent of the perturbationI its dimensions are usually large and it cannot be constructed explicitlyI the response equations are typically solved by iterative techniquesI key step: multiplication of the Hessian with a trial vector

I Analogy with Hooke’s law:

force constant→ kx = −F ← force

I the wave function relaxes by an amount proportional to the perturbation



2n + 1 Rule

I For molecular gradients and Hessians, we have the expressions

dEdx

=∂E

∂x← zero-order response needed

d2Edx2

=∂2E

∂x2+

∂2E

∂x∂λ

∂λ

∂x← first-order response needed

I In general, we have the 2n + 1 rule:

For variational wave functions, the derivatives of the wave function to order ndetermine the derivatives of the energy to order 2n + 1.

I Examples: wave-function responses needed to fourth order:

energy E(0) E(1) E(2) E(3) E(4)

wave function λ(0) λ(0) λ(0), λ(1) λ(0), λ(1) λ(0), λ(1), λ(2)


Analytical Derivative Theory Derivatives for Nonvariational Wave Functions

Nonvariational Wave Functions

I The 2n + 1 rule simplifies property evaluation for variational wave functions

I What about the nonvariational wave functions?

I any energy may be made stationary by Lagrange’s method of undetermined multipliersI the 2n + 1 rule is therefore of general interest

I Example: the CI energy

I the CI energy function is given by:

ECI(x , c, κ) ← CI parameters corbital-rotation parameters κ

I it is nonstationary with respect to the orbital-rotation parameters:

∂ECI(x , c, κ)

∂c= 0 ← stationary

∂ECI(x , c, κ)

∂κ6= 0 ← nonstationary

I We shall now consider its molecular gradient:

1 by straightforward differentiation of the CI energy2 by differentiation of the CI Lagrangian

I In coupled-cluster theory, all parameters are nonvariationally determined


Analytical Derivative Theory Derivatives for Nonvariational Wave Functions

CI Molecular Gradients the Straightforward Way

I Straightforward differentiation of ECI(x , c, κ) gives the expression

dECI

dx=∂ECI

∂x+∂ECI

∂c

∂c

∂x+∂ECI

∂κ

∂κ

∂x

=∂ECI

∂x+∂ECI

∂κ

∂κ

∂x← κ contribution does not vanish

I it appears that we need the first-order response of the orbitals

I The HF orbitals used in CI theory fulfil the following conditions at all geometries:

∂EHF

∂κ= 0 ← HF stationary conditions

I we obtain the orbital responses by differentiating this equation with respect to x :

∂2EHF

∂κ2

∂κ

∂x= −∂

2EHF

∂x∂κ← 1st-order response equations

I one such set of equations must be solved for each perturbation

I Calculated in this manner, the CI gradient becomes expensive


Analytical Derivative Theory Lagrangian Method

Lagrange’s Method of Undetermined Multipliers

I To calculate the CI energy, we minimize ECI with respect to c and κ:

minc,κ

ECI(x , c, κ) subject to the constraints∂EHF(x , κ)

∂κ= 0

I Use Lagrange’s method of undetermined multipliers:

I construct the CI Lagrangian by adding these constraints with multipliers to the energy:

LCI(x , c, κ, κ) = ECI(x , c, κ) + κ

(∂EHF(x , κ)

∂κ− 0

)I adjust the Lagrange multipliers κ such that the Lagrangian becomes stationary:

∂LCI

∂c= 0 =⇒ ∂ECI

∂c= 0 ← CI conditions

∂LCI

∂κ= 0 =⇒ ∂ECI

∂κ+ κ

∂2EHF

∂κ2= 0 ← linear set of equations for κ

∂LCI

∂κ= 0 =⇒ ∂EHF

∂κ= 0 ← HF conditions

I note the duality between κ and κ

I Note that ECI = LCI when the Lagrangian is stationary

I we now have a stationary CI energy expression LCI



CI Molecular Gradients the Easy Way

I The CI Lagrangian is given by

LCI = ECI + κ∂EHF

∂κ← stationary with respect to all variables

I Since the Lagrangian is stationary, we may invoke the 2n + 1 rule:

dECI

dx=

dLCI

dx=∂LCI

∂x=∂ECI

∂x+ κ

∂2EHF

∂κ∂x

zero-order response equations → κ∂2EHF

∂κ2= −∂ECI

∂κ

I This result should be contrasted with the original expression

dECI

dx=∂ECI

∂x+∂ECI

∂κ

∂κ

∂x

first-order response equations →∂2EHF

∂κ2

∂κ

∂x= −∂

2EHF

∂κ∂x

I We have greatly reduced the number of response equations to be solved



Lagrange’s Method Summarized

I (1) Establish the energy function E(x , λ) and identify conditions on the variables

e(x , λ) = 0

I (2) Set up the Lagrangian energy function:

L(x , λ, λ)︸︷︷︸Lagrangian

= E(x , λ)︸︷︷︸energy function

+ λ (e(x , λ)− 0)︸︷︷︸constraints

I (3) Satisfy the stationary conditions for the variables and their multipliers:

∂L

∂λ= e(x , λ) = 0 ← condition for λ determines λ

∂L

∂λ=∂E

∂λ+ λ

∂e

∂λ= 0 ← condition for λ determines λ

I note the duality between λ and λ!

I (4) Calculate derivatives from the stationary Lagrangian

I The Lagrangian approach is generally applicable:I it gives the Hylleraas functional when applied to a perturbation expressionI it may be generalized to time-dependent properties



2n + 1 and 2n + 2 Rules

I For variational wave functions, we have the 2n + 1 rule:

λ(n) determines the energy to order 2n + 1.

I The Lagrangian technique extends this rule to nonvariational wave functions

I For the new variables—the multipliers—the stronger 2n + 2 rule applies:

λ(n)

determines the energy to order 2n + 2.

I Responses required to order 10:

E(n) 0 1 2 3 4 5 6 7 8 9 10

λ(k) 0 0 1 1 2 2 3 3 4 4 5

λ(k)

0 0 0 1 1 2 2 3 3 4 4


Analytical Derivative Theory Hartree–Fock Molecular Gradient

Hartree–Fock Energy

I The MOs are expanded in atom-fixed AOs

φp(r; x) =∑

µCµpχµ(r; x)

I The HF energy may be written in the general form

EHF =∑

pqDpqhpq + 1

2

∑pqrs

dpqrsgpqrs +∑

K>L

ZKZL

RKL

wherehpq(x) =

∫φp(r, x)

(− 1

2∇2 −∑K

ZKrK

)φq(r, x)dr

gpqrs(x) =∫∫ φp(r1,x)φq(r1,x)φr (r2,x)φs (r2,x)

r12dr1dr2

I the integrals depend explicitly on the geometry

I In closed-shell restricted HF (RHF) theory, the energy is given by

ERHF = 2∑

ihii +

∑ij

(2giijj − gijji

)+∑

K>L

ZKZL

RKL

I summations over doubly occupied orbitals



Hartree–Fock Equations

I The HF energy is minimized subject to orthonormality constraints

Sij =⟨φi |φj

⟩= δij

I We therefore introduce the HF Lagrangian:

LHF = EHF −∑

ijεij(Sij − δij

)=∑

ijDijhij + 1

2

∑ijkl

dijklgijkl −∑

ijεij(Sij − δij

)+∑

K>L

ZKZL

RKL

I The stationary conditions on the Lagrangian become:

∂LHF

∂εij= Sij − δij = 0

∂LHF

∂Cµi=∂EHF

∂Cµi−∑

klεkl

∂Skl

∂Cµi= 0

I the multiplier conditions are the orthonormality constraintsI the MO stationary conditions are the Roothaan–Hall equations

∂EHF

∂Cµi=∑

klεkl

∂Skl

∂Cµi=⇒ FAOC = SAOCε



Hartree–Fock Molecular Gradient

I From the 2n + 1 rule, we obtain the RHF molecular gradient:

dEHF

dx=

dLHF

dx=∂LHF

∂x=∂EHF

∂x−∑ij

εij∂Sij

∂x

I In terms of MO integrals and density-matrix elements, we obtain the expression

dEHF

dx=∑ij

Dij∂hij

∂x+

1

2

∑ijkl

dijkl∂gijkl

∂x−∑ij

εij∂Sij

∂x+ Fnuc

I We then transform to the AO basis:

dEHF

dx=∑µν

DAOµν

∂hAOµν

∂x+

1

2

∑µνρσ

dAOµνρσ

∂gAOµνρσ

∂x−∑µν

εAOµν

∂SAOµν

∂x+ Fnuc

I density matrices transformed to AO basisI derivative integrals added directly to gradient elements

I Important points:

I the gradient does not involve MO differentiation because of the 2n + 1 ruleI the time-consuming step is integral differentiation


Analytical Derivative Theory FCI Lagrangian and Derivatives

FCI Energy

I Consider a normalized CI wave function:

|c〉 =∑∞

n=0cn|n〉, cTc = 1, 〈m|n〉 = δmn

I The basis functions |n〉 are the normalized CI eigenstates of the unperturbed problem:

〈m|H|n〉 = δmnEn, E0 ≤ E1 ≤ E2 · · ·I We assume that the ground-state energy function depends on two external parameters:

〈c|H(x , y)|c〉 =∑

mncm〈m|H(x , y)|n〉cn,

∑n

c2n = 1

I We construct a variational CI Lagrangian:

L(x , y , c, µ) =∑mn

cm〈m|H(x , y)|n〉cn − µ(∑

n

c2n − 1

)I The stationary conditions are given by

∂L(x , y , c, µ)

∂cn= 0 =⇒ 2〈n|H(x , y)|c〉 − 2µcn = 0 =⇒ H(x , y)c = E0(x , y)c

∂L(x , y , c, µ)

∂µ= 0 =⇒

∑n

c2n − 1 = 0 =⇒ cTc = 1

I the first condition is the CI eigenvalue problem with ground-state energy E0(x, y) = µI the second condition is the CI normalization condition


Analytical Derivative Theory FCI Lagrangian and Derivatives

FCI Molecular Gradient and Hessian

I Using the CI Lagrangian, we calculate CI energy derivative in the usual way:

dE

dx=∂L

∂x,

d2E

dxdy=

∂2L

∂x∂y+∑n

∂2L

∂x∂cn

∂cn

∂x,∑n

∂2L

∂cm∂cn

∂cn

∂x= − ∂2L

∂x∂cm

I By inverting the electronic Hessian, we obtain the more compact expression:

d2E

dxdy=

∂2L

∂x∂y−∑mn

∂2L

∂x∂cm

[∂2L

∂cm∂cn

]−1∂2L

∂cn∂y

I We next evaluate the various partial derivatives at x = y = 0 where |c〉 = |0〉:

∂L

∂x=⟨

0∣∣∣ ∂H∂x ∣∣∣ 0

⟩,

∂2L

∂x∂y=⟨

0∣∣∣ ∂2H∂x∂y

∣∣∣ 0⟩,

∂2L

∂x∂cn= 2

⟨n∣∣∣ ∂H∂x ∣∣∣ 0

⟩∂2L

∂cm∂cn= 2 〈m |H − E0| n〉 = 2(En − E0)δmn

I Inserted above, we recover Rayleigh–Schrodinger perturbation theory to second order:

dE

dx=⟨

0∣∣∣ ∂H∂x ∣∣∣ 0

⟩,

d2E

dxdy=⟨

0∣∣∣ ∂2H∂x∂y

∣∣∣ 0⟩− 2

∑n

⟨0∣∣∣ ∂H∂x ∣∣∣ n⟩⟨n ∣∣∣ ∂H∂y ∣∣∣ 0

⟩En − E0


Analytical Derivative Theory Second-Quantization Hamiltonian for Derivatives

Second-Quantization Hamiltonian

I In second quantization, the Hamiltonian operator is given by:

H =∑pq

hpqa†paq +

∑pqrs

gpqrsa†pa†r asaq + hnuc

hpq = 〈φ∗p (r)|h(r)|φq(r)〉gpqrs = 〈φ∗p (r1)φ∗r (r2)|r−1

12 |φq(r1)φs(r2)〉

I Its construction assumes an orthonormal basis of MOs φp :

[ap , aq ]+ = 0, [a†p , a†q ]+ = 0, [ap , a

†q ]+ = δpq

I The MOs are expanded in AOs, which often depend explicitly on the perturbation

I such basis sets are said to be perturbation-dependent:

φp(r) =∑µ

Cpµ χµ(r, x)

I we must make sure that the MOs remain orthonormal for all xI this introduces complications as we take derivatives with respect to x



MOs and Hamiltonian at Distorted geometries

1. Orthonormal MOs at the reference geometry:

φ(x0) = C(0)χ(x0)

S(x0) =⟨φ(x0) |φ†(x0)

⟩= I

2. Geometrical distortion x = x0 + ∆x :

φ(x) = C(0)χ(x)

S(x) =⟨φ(x) |φ†(x)

⟩6= I

note: this basis is nonorthogonal and not useful for setting up the Hamiltonian.

3. Orthonormalize the basis set (e.g., by Lowdin orthonormalization):

ψ(x) = S−1/2(x)φ(x)

S(x) = S−1/2(x)S(x)S−1/2(x) = I

4. From these orthonormalized MOs (OMOs) ψp , construct Hamiltonian in the usual manner

H =∑

pqhpqa

†paq +

∑pqrs

gpqrsa†pa†r asaq + hnuc

hpq = 〈ψ∗p (r)|h(r)|ψq(r)〉gpqrs = 〈ψ∗p (r1)ψ∗r (r2)|r−1

12 |ψq(r1)ψs(r2)〉



Hamiltonian at all Geometries

I The Hamiltonian is now well defined at all geometries:

H(x) =∑pq

hpq(x)Epq(x) + 12

∑pqrs

gpqrs(x)epqrs(x) + hnuc(x)

I The OMO integrals are given by

hpq(x) =∑mn

hmn(x)[S−1/2]mp(x)[S−1/2]nq(x)

in terms of the usual MO integrals

hmn(x) =∑µν

C(0)mµC

(0)nν h

AOµν (x), Smn(x) =

∑µν

C(0)mµC

(0)nν S

AOµν (x)

and similarly for the two-electron integrals.

I What about the geometry dependence of the excitation operators?

I this may be neglected when calculating derivatives since, for all geometries,[ap(x), a†q(x)

]+

= Spq(x) = δpq



HF Molecular Gradients in Second Quantization

I The molecular gradient now follows from the Hellmann–Feynman theorem:

E (1) = 〈0|H(1)|0〉 =∑pq

Dpq h(1)pq + 1

2

∑pqrs

dpqrs g(1)pqrs + h

(1)nuc

I We need the derivatives of the OMO integrals:

h(1)pq =

∑mn

[hmn(S−1/2)mp(S−1/2)nq

](1)= h

(1)pq − 1

2

∑m

S(1)pmh

(0)mq − 1

2

∑m

h(0)pmS

(1)mq

I The gradient may therefore be written in the form

E (1) =∑pq

Dpqh(1)pq + 1

2

∑pqrs

dpqrsg(1)pqrs −

∑pq

FpqS(1)pq + h

(1)nuc,

where the generalized Fock matrix is given by:

Fpq =∑n

Dpnhqn +∑nrs

dpnrsgqnrs

I For RHF theory, this result is equivalent to that derived in first quantization


Geometrical Properties

Table of Contents









Geometrical Derivatives

I In the Born–Oppenheimer approximation, the nuclei move on the electronic potential-energysurface E(x), which is a function of the nuclear geometry:

E(x) = E0 + E(1)∆x + 12E(2)∆x2 + · · · ← expansion around the reference geometry

I The derivatives of this surface are therefore important:

E(1) =dEdx

← molecular gradient

E(2) =d2Edx2

← molecular Hessian

I The geometrical derivatives are

I used for locating and characterizing critical pointsI related to spectroscopic constants, vibrational frequencies, and intensities

I Usually, only a few terms are needed in the expansions

I in some cases low-order expansions are inadequate or useless



Uses of Geometrical Derivatives

I To explore molecular potential-energy surfaces (3N − 6 dimensions)

I localization and characterization of stationary pointsI localization of avoided crossings and conical intersectionsI calculation of reaction paths and reaction-path HamiltoniansI application to direct dynamics

I To calculate spectroscopic constants

I molecular structureI quadratic force constants and harmonic frequenciesI cubic and quartic force constants; fundamental frequenciesI partition functionsI dipole gradients and vibrational infrared intensitiesI polarizability gradients and Raman intensities


Geometrical Properties Bond Distances

Bond Distances

I Mean and mean abs. errors for 28 distances at the all-el. cc-pVXZ level (pm):

-2

-1

1

HF

MP2

MP3

MP4

CCSD

CCSD(T)

CISD

pVDZ

pVTZpVQZ

|∆| DZ TZ QZCCSD 1.2 0.6 0.8CCSD(T) 1.7 0.2 0.2

I Bonds shorten with increasing basis:

I HF: DZ → TZ 0.8 pm; TZ → QZ 0.1 pmI corr.: DZ → TZ 1.6 pm; TZ → QZ 0.1–0.2 pm

I Bonds lengthen with improvements in the N-electron model:

I singles < doubles < triples < · · ·I There is considerable scope for error cancellation: CISD/DZ, MP3/DZ



Bond Distances Re of BH, CO, N2, HF, and F2 (pm)

-6 6

SCFcc-pCVDZ 2.5

-6 6 -6 6

SCFcc-pCVTZ 3.4

-6 6 -6 6

SCFcc-pCVQZ 3.5

-6 6 -6 6

SCFcc-pCV5Z 3.5

-6 6

-6 6

MP2cc-pCVDZ 1.5

-6 6 -6 6

MP2cc-pCVTZ 0.9

-6 6 -6 6

MP2cc-pCVQZ 0.8

-6 6 -6 6

MP2cc-pCV5Z 0.8

-6 6

-6 6

CCSDcc-pCVDZ 1.2

-6 6 -6 6

CCSDcc-pCVTZ 0.6

-6 6 -6 6

CCSDcc-pCVQZ 0.9

-6 6 -6 6

CCSDcc-pCV5Z 1.

-6 6

-6 6

CCSD(T)cc-pCVDZ 2.1

-6 6 -6 6

CCSD(T)cc-pCVTZ 0.2

-6 6 -6 6

CCSD(T)cc-pCVQZ 0.1

-6 6 -6 6

CCSD(T)cc-pCV5Z 0.1

-6 6



Contributions to Equilibrium Bond Distances (pm)

RHF SD T Q 5 rel. adia. theory exp. err.HF 89.70 1.67 0.29 0.02 0.00 0.01 0.0 91.69 91.69 0.00N2 106.54 2.40 0.67 0.14 0.03 0.00 0.0 109.78 109.77 0.01F2 132.64 6.04 2.02 0.44 0.03 0.05 0.0 141.22 141.27 −0.05CO 110.18 1.87 0.75 0.04 0.00 0.00 0.0 112.84 112.84 0.00

I We have agreement with experiment to within 0.01 pm except for F2

I Hartree–Fock theory underestimates bond distances by up to 8.6 pm (for F2)

I All correlation contributions are positive

I approximately linear convergence, slowest for F2I triples contribute up to 2.0, quadruples up to 0.4, and quintuples 0.03 pmI sextuples are needed for convergence to within 0.01 pm

I Relativistic corrections are small except for F2 (0.05 pm)

I of the same magnitude and direction as the quintuples


Geometrical Properties Harmonic and Anharmonic Constants

Harmonic Constants ωe of BH, CO, N2, HF, and F2 (cm−1)

-250 250

SCFcc-pCVDZ 269

-250 250 -250 250

SCFcc-pCVTZ 288

-250 250 -250 250

SCFcc-pCVQZ 287

-250 250 -250 250

SCFcc-pCV5Z 287

-250 250

-250 250

MP2cc-pCVDZ 68

-250 250 -250 250

MP2cc-pCVTZ 81

-250 250 -250 250

MP2cc-pCVQZ 73

-250 250 -250 250

MP2cc-pCV5Z 71

-250 250

-250 250

CCSDcc-pCVDZ 34

-250 250 -250 250

CCSDcc-pCVTZ 64

-250 250 -250 250

CCSDcc-pCVQZ 71

-250 250 -250 250

CCSDcc-pCV5Z 72

-250 250

-250 250

CCSD(T)cc-pCVDZ 42

-250 250 -250 250

CCSD(T)cc-pCVTZ 14

-250 250 -250 250

CCSD(T)cc-pCVQZ 9

-250 250 -250 250

CCSD(T)cc-pCV5Z 10

-250 250



Contributions to Harmonic Frequencies ωe (cm−1)

RHF SD T Q 5 rel. adia. theory exp. err.HF 4473.8 −277.4 −50.2 −4.1 −0.1 −3.5 0.4 4138.9 4138.3 0.1N2 2730.3 −275.8 −72.4 −18.8 −3.9 −1.4 0.0 2358.0 2358.6 −0.6F2 1266.9 −236.1 −95.3 −15.3 −0.8 −0.5 0.0 918.9 916.6 2.3CO 2426.7 −177.4 −71.7 −7.2 0.0 −1.3 0.0 2169.1 2169.8 0.7

I We have agreement with experiment to within 1 cm−1 except for F2

I Hartree–Fock theory overestimates harmonic frequencies by up to 38% (in F2).

I All correlation contributions are large and negative

I triples contribute up to 95 cm−1, quadruples 20 cm−1, and quintuples 4 cm−1

I sextuples are sometimes needed for convergence to within 1 cm−1

I The relativistic corrections are of the order of 1 cm−1

I of the same magnitude and direction as the quadruples or quintuples



Higher-Order Connected Contributions to ωe in N2 (cm−1)

I There are substantial higher-order corrections:

371.9

84.6

4.113.8

23.54.7 0.8

HF CCSD!FC CCSD"T#!FC CCSD"T# CCSDT CCSDTQ CCSDTQ5

I connected triples relaxation contributes 9.7 cm−1 (total triples −70.5 cm−1)I connected quadruples contribute −18.8 cm−1

I connected quintuples contribute −3.9 cm−1



Anharmonic Constants ωexe of BH, CO, N2, HF, and F2 (cm−1)

-6 6

SCFcc-pCVDZ 4

-6 6 -6 6

SCFcc-pCVTZ 5

-6 6 -6 6

SCFcc-pCVQZ 4

-6 6 -6 6

SCFcc-pCV5Z 4

-6 6

-6 6

MP2cc-pCVDZ 3

-6 6 -6 6

MP2cc-pCVTZ 3

-6 6 -6 6

MP2cc-pCVQZ 3

-6 6 -6 6

MP2cc-pCV5Z 3

-6 6

-6 6

CCSDcc-pCVDZ 2

-6 6 -6 6

CCSDcc-pCVTZ 2

-6 6 -6 6

CCSDcc-pCVQZ 2

-6 6 -6 6

CCSDcc-pCV5Z 1

-6 6

-6 6

CCSD(T)cc-pCVDZ 1

-6 6 -6 6

CCSD(T)cc-pCVTZ 1

-6 6 -6 6

CCSD(T)cc-pCVQZ 0

-6 6 -6 6

CCSD(T)cc-pCV5Z 0

-6 6


Electronic Hamiltonian

Table of Contents








Electronic Hamiltonian Particle in a Conservative Force Field

Hamiltonian Mechanics

I In classical Hamiltonian mechanics, a system of particles is described in terms theirpositions qi and conjugate momenta pi .

I For each such system, there exists a scalar Hamiltonian function H(qi , pi ) such that theclassical equations of motion are given by:

qi =∂H

∂pi, pi = − ∂H

∂qi(Hamilton’s equations of motion)

I Example: a single particle of mass m in a conservative force field F (q)

I the Hamiltonian function is constructed from the corresponding scalar potential:

H(q, p) =p2

2m+ V (q), F (q) = −∂V (q)

∂q

I Hamilton’s equations are equivalent to Newton’s equations:

q = ∂H(q,p)∂p

= pm

p = − ∂H(q,p)∂q

= − ∂V (q)∂q

=⇒ mq = F (q) (Newton’s equations of motion)

I Note:

I Newton’s equations are second-order differential equationsI Hamilton’s equations are first-order differential equationsI the Hamiltonian function is not unique!


Electronic Hamiltonian Particle in a Conservative Force Field

Quantization of a Particle in a Conservative Force Field

I The Hamiltonian formulation is more general than the Newtonian formulation:

I it is invariant to coordinate transformationsI it provides a uniform description of matter and fieldI it constitutes the springboard to quantum mechanics

I The Hamiltonian function (the total energy) of a particle in a conservative force field:

H(q, p) =p2

2m+ V (q)

I Standard rule for quantization (in Cartesian coordinates):

I carry out the substitutions

p→ −i~∇, H → i~∂

∂t

I multiply the resulting expression by the wave function Ψ(q) from the right:

i~∂Ψ(q)

∂t=

[− ~2

2m∇2 + V (q)

]Ψ(q)

I This approach is sufficient for a treatment of electrons in an electrostatic field

I it is insufficient for nonconservative systemsI it is therefore inappropriate for systems in a general electromagnetic field


Electronic Hamiltonian Particle in a Lorentz Force Field

Lorentz Force and Maxwell’s Equations

I In the presence of an electric field E and a magnetic field (magnetic induction) B,a classical particle of charge z experiences the Lorentz force:

F = z (E + v × B)

I since this force depends on the velocity v of the particle, it is not conservative

I The electric and magnetic fields E and B satisfy Maxwell’s equations (1861–1868):

∇ · E = ρ/ε0 ← Coulomb’s law

∇× B− ε0µ0 ∂E/∂t = µ0J ← Ampere’s law with Maxwell’s correction

∇ · B = 0

∇× E + ∂B/∂t = 0 ← Faraday’s law of induction

I Note:

I when the charge and current densities ρ(r, t) and J(r, t) are known,Maxwell’s equations can be solved for E(r, t) and B(r, t)

I on the other hand, since the charges (particles) are driven by the Lorentz force,ρ(r, t) and J(r, t) are functions of E(r, t) and B(r, t)

I We here consider the motion of particles in a given (fixed) electromagnetic field



Scalar and Vector Potentials

I The second, homogeneous pair of Maxwell’s equations involves only E and B:

∇ · B = 0 (1)

∇× E +∂B

∂t= 0 (2)

1 Eq. (1) is satisfied by introducing the vector potential A:

∇ · B = 0 =⇒ B = ∇× A ← vector potential (3)

2 inserting Eq. (3) in Eq. (2) and introducing a scalar potential φ, we obtain

∇×(

E +∂A

∂t

)= 0 =⇒ E +

∂A

∂t= −∇φ ← scalar potential

I The second pair of Maxwell’s equations is thus automatically satisfied by writing

E = −∇φ−∂A

∂t

B = ∇× A

I The potentials (φ,A) contain four rather than six components as in (E,B).

I They are obtained by solving the first, inhomogeneous pair of Maxwell’s equations,which contains ρ and J.



Gauge Transformations

I The scalar and vector potentials φ and A are not unique.

I Consider the following transformation of the potentials:

φ′ = φ− ∂f∂t

A′ = A + ∇f

f = f (q, t) ← gauge function of position and time

I This gauge transformation of the potentials does not affect the physical fields:

E′ = −∇φ′ −

∂A′

∂t= −∇φ + ∇

∂f

∂t−∂A

∂t−∂∇f

∂t= E

B′ = ∇× A′ = ∇× (A + ∇f ) = B + ∇×∇f = B

I We are free to choose f (q, t) to make the potentials satisfy additional conditions

I Typically, we require the vector potential to be divergenceless:

∇ · A′ = 0 =⇒ ∇ · (A + ∇f ) = 0 =⇒ ∇2f = −∇ · A ← Coulomb gauge

I We shall always assume that the vector potential satisfies the Coulomb gauge:

∇× A = B, ∇ · A = 0 ← Coulomb gauge

I Note: A is still not uniquely determined, the following transformation being allowed:

A′ = A + ∇f , ∇2f = 0



Hamiltonian in an Electromagnetic Field

I We must construct a Hamiltonian function such thatHamilton’s equations are equivalent to Newton’s equation with the Lorentz force:

qi =∂H

∂pi& pi = − ∂H

∂qi⇐⇒ ma = z (E + v × B)

I To this end, we introduce scalar and vector potentials φ and A such that

E = −∇φ− ∂A

∂t, B = ∇× A

I In terms of these potentials, the classical Hamiltonian function becomes

H =π2

2m+ zφ, π = p− zA ← kinetic momentum

I Quantization is then accomplished in the usual manner, by the substitutions

p→ −i~∇, H → i~∂

∂t

I The time-dependent Schrodinger equation for a particle in an electromagnetic field:

i~∂Ψ

∂t=

1

2m(−i~∇− zA) · (−i~∇− zA) Ψ + zφΨ


Electronic Hamiltonian Electron Spin

Electron Spin

I The nonrelativistic Hamiltonian for an electron in an electromagnetic field is then given by:

H =π2

2m− eφ, π = −i~∇ + eA

I However, this description ignores a fundamental property of the electron: spin.

I Spin was introduced by Pauli in 1927, to fit experimental observations:

H =(σ · π)2

2m− eφ =

π2

2m+

e~2m

B · σ − eφ

where σ contains three operators, represented by the two-by-two Pauli spin matrices

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

)I The Schrodinger equation now becomes a two-component equation:(

π2

2m− eφ+ e~

2mBz

e~2m

(Bx − iBy )e~2m

(Bx + iBy ) π2

2m− eφ− e~

2mBz

)(ΨαΨβ

)= E

(ΨαΨβ

)

I Note: the two components are only coupled in the presence of an external magnetic field


Electronic Hamiltonian Electron Spin

Spin and Relativity

I The introduction of spin by Pauli in 1927 may appear somewhat ad hoc

I By contrast, spin arises naturally from Dirac’s relativistic treatment in 1928

I is spin a relativistic effect?

I However, reduction of Dirac’s equation to nonrelativistic form yields the Hamiltonian

H =(σ · π)2

2m− eφ =

π2

2m+

e~2m

B · σ − eφ 6= π2

2m− eφ

I in this sense, spin is not a relativistic property of the electronI but we note that, in the nonrelativistic limit, all magnetic fields disappear. . .

I We interpret σ by associating an intrinsic angular momentum (spin) with the electron:

s = ~σ/2


Electronic Hamiltonian Molecular Electronic Hamiltonian

Molecular Electronic Hamiltonian

I The nonrelativistic Hamiltonian for an electron in an electromagnetic field is therefore

H =π2

2m+

e

mB · s− eφ, π = p + eA, p = −i~∇

I expanding π2 and assuming the Coulomb gauge ∇ · A = 0, we obtain

π2Ψ = (p + eA) · (p + eA) Ψ = p2Ψ + ep · AΨ + eA · pΨ + e2A2Ψ

= p2Ψ + e(p · A)Ψ + 2eA · pΨ + e2A2Ψ =(p2 + 2eA · p + e2A2

)Ψ

I in molecules, the dominant electromagnetic contribution is from the nuclear charges:

φ = − 14πε0

∑K

ZK erK

+ φext

I Summing over all electrons and adding pairwise Coulomb interactions, we obtain

H =∑i

1

2mp2i −

e2

4πε0

∑Ki

ZK

riK+

e2

4πε0

∑i>j

r−1ij ← zero-order Hamiltonian

+e

m

∑i

Ai · pi +e

m

∑i

Bi · si − e∑i

φi ← first-order Hamiltonian

+e2

2m

∑i

A2i ← second-order Hamiltonian


Electronic Hamiltonian Molecular Electronic Hamiltonian

Magnetic Perturbations

I In atomic units, the molecular Hamiltonian is given by

H = H0 +∑i

A(ri ) · pi︸︷︷︸orbital paramagnetic

+∑i

B(ri ) · si︸︷︷︸spin paramagnetic

−∑i

φ(ri ) +1

2

∑i

A2(ri )︸︷︷︸diamagnetic

I There are two kinds of magnetic perturbation operators:

I the paramagnetic operator is linear and may lower or raise the energyI the diamagnetic operator is quadratic and always raises the energy

I There are two kinds of paramagnetic operators:

I the orbital paramagnetic operator couples the field to the electron’s orbital motionI the spin paramagnetic operator couples the field to the electron’s spin

I In the study of magnetic properties, we are interested in two types of perturbations:

I uniform external magnetic field B, with vector potential

Aext(r) =1

2B× r leads to Zeeman interactions

I nuclear magnetic moments MK , with vector potential

Anuc(r) = α2∑K

MK × rKr3K

leads to hyperfine interactions

where α ≈ 1/137 is the fine-structure constant


London Orbitals

Table of Contents








London Orbitals

Hamiltonian in a Uniform Magnetic Field

I The nonrelativistic electronic Hamiltonian (implied summation over electrons):

H = H0 + A (r) · p + B (r) · s + 12A (r)2

I The vector potential of the uniform field B is given by:

B = ∇× A = const =⇒ AO(r) = 12

B× (r −O) = 12

B× rO

I note: the gauge origin O is arbitrary!

I The orbital paramagnetic interaction:

AO(r) · p = 12

B× (r −O) · p = 12

B · (r −O)× p = 12

B · LO

where we have introduced the angular momentum relative to the gauge origin:

LO = rO × p

I The diamagnetic interaction:

12A2

O (r) = 18

(B× rO) · (B× rO) = 18

[B2r2

O − (B · rO)2]

I The electronic Hamiltonian in a uniform magnetic field depends on the gauge origin:

H = H0 +1

2B · LO + B · s +

1

8

[B2r2

O − (B · rO)2]

I a change of the origin is a gauge transformation


London Orbitals Gauge-Origin Transformations

Gauge Transformation of Schrodinger Equation

I What is the effect of a gauge transformation on the wave function?

I Consider a general gauge transformation for the electron (atomic units):

A′ = A + ∇f , φ′ = φ− ∂f

∂t

I It can be shown this represents a unitary transformation of H − i∂/∂t:(H′ − i

∂

∂t

)= exp (−if )

(H − i

∂

∂t

)exp (if )

I In order that the Schrodinger equation is still satisfied(H′ − i

∂

∂t

)Ψ′ ⇐⇒

(H − i

∂

∂t

)Ψ,

the new wave function must undergo a compensating unitary transformation:

Ψ′ = exp (−if ) Ψ

I All observable properties such as the electron density are then unaffected:

ρ′ = (Ψ′)∗Ψ′ = [Ψ exp(−if )]∗[exp(−if )Ψ] = Ψ∗Ψ = ρ


London Orbitals Gauge-Origin Transformations

Gauge-Origin Transformations

I Different choices of gauge origin in the external vector potential

AO (r) = 12

B× (r −O)

are related by gauge transformations:

AG (r) = AO (r)− AO (G) = AO (r) + ∇f , f (r) = −AO (G) · r

I The exact wave function transforms accordingly and gives gauge-invariant results:

ΨexactG = exp [−if (r)] Ψexact

O = exp [iAO (G) · r] ΨexactO (rapid) oscillations

I Illustration: H2 on the z axis in a magnetic field B = 0.2 a.u. in the y direction

I wave function with gauge origin at O = (0, 0, 0) (left) and G = (100, 0, 0) (right)

London orbitals: do we need them?

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

A = 120 z r ! A0 = A + r(A(G) · r)

= RHF/aug-cc-pVQZ ! 0 = eiA(G)·r (10)

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

Space coordinate, x (along the bond)

Wav

e fun

ction

, 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

Space coordinate, x (along the bond)

Gaug

e tra

nsfor

med w

ave f

uncti

on, ψ"

Re(ψ")Im(ψ")|ψ"|2

Erik Tellgren Ab initio finite magnetic field calculations using London orbitalsTrygve Helgaker (CTCC, University of Oslo) Derivatives and Properties MWM Summer School 2016 54 / 87

London Orbitals London Orbitals

London Orbitals

I The exact wave function transforms in the following manner:

ΨexactG = exp

[i 1

2B× (G−O) · r

]Ψexact

O

I this behaviour cannot easily be modelled by standard atomic orbitals

I Let us build this behaviour directly into the atomic orbitals:

ωlm(rK ,B,G) = exp[i 1

2B× (G− K) · r

]χlm(rK)

I χlm(rK ) is a normal atomic orbital centred at K and quantum numbers lmI ωlm(rK ,B,G) is a field-dependent orbital at K with field B and gauge origin G

I Each AO now responds in a physically sound manner to an applied magnetic field

I indeed, all AOs are now correct to first order in B, for any gauge origin GI the calculations become rigorously gauge-origin independentI uniform (good) quality follows, independent of molecule size

I These are the London orbitals after Fritz London (1937)

I also known as GIAOs (gauge-origin independent AOs or gauge-origin including AOs)

I Questions:

I are London orbitals needed in atoms?I why not attach the phase factor to the total wave function instead?


London Orbitals London Orbitals

Dissociation With and Without London Orbitals

I Let us consider the FCI dissociation of H2 in a magnetic field

I full lines: with London atomic orbitalsI dashed lines: without London atomic orbitals

2 4 6 8 10

-1.0

-0.5

0.5

1.0

B = 0.0

B = 1.0

B = 2.5

I Without London orbitals, the FCI method is not size extensive in magnetic fields


Zeeman and Hyperfine Interactions

Table of Contents









Hamiltonian in Magnetic Field

I In atomic units, the molecular Hamiltonian is given by

H = H0 + A(r) · p︸︷︷︸orbital paramagnetic

+ B(r) · s︸︷︷︸spin paramagnetic

+ 12A2(r)︸︷︷︸

diamagnetic

I There are two kinds of magnetic perturbation operators:

I paramagnetic (may lower or raise energy) and diamagnetic (always raises energy)

I There are two kinds of paramagnetic operators:

I the orbital paramagnetic and spin paramagnetic

I First- and second-order Rayleigh–Schrodinger perturbation theory gives:

E (1) = 〈0 |A · p + B · s| 0〉

E (2) =1

2

⟨0∣∣A2∣∣ 0⟩−∑n

〈0 |A · p + B · s| n〉〈n |A · p + B · s| 0〉En − E0

I In the study of magnetic properties, we are interested in two types of perturbations:

I externally applied uniform magnetic fields BI fields generated internally by nuclear magnetic moments MK

I Both fields are weak—well described by perturbation (response) theory


Zeeman and Hyperfine Interactions Paramagnetic Operators

Orbital Paramagnetic Interactions: A · p

I Vector potentials corresponding to uniform fields and nuclear magnetic moments:

AO =1

2B× rO, AK = α2 MK × rK

r3K

, α ≈ 1/137

I the external field is typically about 10−4 a.u. (NMR experiments)I the nuclear vector potential is exceedingly small (about 10−8 a.u.) since:

α2 = c−2 ≈ 10−4 a.u., MK = γK~IK ≈ 10−4 a.u.

I We obtain the following orbital paramagnetic operators:

AO · p =1

2B× rO · p =

1

2B · rO × p =

1

2B · LO ← orbital Zeeman

AK · p = α2 MK × rK · p

r3K

= α2 MK · rK × p

r3K

= α2MK ·

LK

r3K

← orbital hyperfine

I interactions depend on angular momenta LO and LK relative to O and RK , respectivelyI orbital hyperfine interaction expressed in terms of the paramagnetic spin–orbit operator:

AK · p = MK · hPSOK , hPSO

K = α2 LK

r3K

I These are imaginary singlet operators

I they have zero expectation values of closed-shell statesI they generate complex wave functions


Zeeman and Hyperfine Interactions Paramagnetic Operators

Spin Paramagnetic Interactions: B · s

I The spin interaction with the external uniform field B is trivial:

B · s ← spin Zeeman interaction

I should be compared with the orbital Zeeman interaction 12 B · LO (different prefactor!)

I Taking the curl of AK , we obtain the nuclear magnetic field:

BK = ∇× AK =8πα2

3δ(rK )MK + α2 3rK (rK ·MK )− r2

KMK

r5K

I the first term is a contact interaction and contributes only at the nucleusI the second term is a classical dipole field and contributes at a distance

I This magnetic field BK thus gives rise to two spin hyperfine interactions:

BK · s = MK · (hFCK + hSD

K ),

hFCK = 8πα2

3δ(rK ) s Fermi contact (FC)

hSDK = α2 3rK rTK−r2

K I3

r5K

s spin–dipole (SD)

I the FC operator contributes when the electron passes through the nucleusI the SD operator is a classical dipolar interaction, decaying as r−3

K

I These are real triplet operators, which change the spin of the wave function

I they have zero expectation values of closed-shell statesI they couple closed-shell states to triplet states


Zeeman and Hyperfine Interactions Hamiltonian with Zeeman and Hyperfine Operators

Perturbation Theory with Zeeman and Hyperfine Operators

I Hamiltonian with a uniform external field and with nuclear magnetic moments:

H = H0 + H(1) + H(2) = H0 + H(1)z + H

(1)hf + 1

2A2

I Zeeman interactions with the external magnetic field B:

H(1)z =

1

2B · LO + B · s ∼ 10−4

I hyperfine interactions with the nuclear magnetic moments MK :

H(1)hf =

∑K

MK · hPSOK +

∑K

MK ·(

hFCK + hSD

K

)∼ 10−8

hPSOK = α2 LK

r3K

, hFCK =

8πα2

3δ(rK ) s, hSD

K = α2 3rK rTK − r2K I3

r5K

s

I Second-order Rayleigh–Schrodinger perturbation theory:

E (1) =⟨0∣∣H(1)

z + H(1)hf

∣∣0⟩E (2) =

1

2

⟨0∣∣A2∣∣ 0⟩−∑n

⟨0∣∣H(1)

z + H(1)hf

∣∣n⟩⟨n∣∣H(1)z + H

(1)hf

∣∣0⟩En − E0


Zeeman and Hyperfine Interactions Hamiltonian with Zeeman and Hyperfine Operators


PSO hyperfine

Zeeman

PSOFC+SDFC+SD

SS, SO, OO

SO SO


Zeeman and Hyperfine Interactions Diamagnetic Operators

Diamagnetic Operators: 12A

2

I From AO and AK , we obtain three diamagnetic operators:

A = AO + AK =⇒ A2 = AO · AO + 2AO · AK + AK · AK

I Their explicit forms and typical magnitudes (atomic units) are given by

AO · AO =1

4(B× rO) · (B× rO) ∼ 10−8

AO · AK =α2

2

(B× rO) · (MK × rK )

r3K

∼ 10−12

AK · AL = α4 (MK × rK ) · (ML × rL)

r3K r

3L

∼ 10−16

I These are all real singlet operators

I their expectation values contribute to second-order magnetic propertiesI they are all exceedingly small but nonetheless all observable


Molecular Magnetic Properties

Table of Contents








Molecular Magnetic Properties

Taylor Expansion of Energy

I Expand the energy in the presence of an external magnetic field B and nuclear magneticmoments MK around zero field and zero moments:

E (B,M) = E0 +

perm. magnetic moments︷︸︸︷BTE(10) +

hyperfine coupling︷︸︸︷∑K

MTK E

(01)K

+1

2BTE(20)B︸︷︷︸− magnetizability

+1

2

∑K

BTE(11)K MK︸︷︷︸

shieldings + 1

+1

2

∑KL

MTK E

(02)KL ML︸︷︷︸

spin–spin couplings

+ · · ·

I First-order terms vanish for closed-shell systems because of symmetry

I they shall be considered only briefly here

I Second-order terms are important for many molecular properties

I magnetizabilitiesI nuclear shieldings constants of NMRI nuclear spin–spin coupling constants of NMRI electronic g tensors of EPR (not dealt with here)

I Higher-order terms are negligible since the perturbations are tiny:

1) the magnetic induction B is weak (≈ 10−4 a.u.)2) the nuclear magnetic moments MK couple weakly (µ0µN ≈ 10−8 a.u.)


Molecular Magnetic Properties First-Order Magnetic Properties

First-Order Molecular Properties

I The first-order properties are expectation values of H(1)

I Permanent magnetic moment

M =⟨0∣∣H(1)

z

∣∣0⟩ =⟨0∣∣ 1

2LO + s

∣∣0⟩I permanent magnetic moment dominates the magnetism of moleculesI the molecule reorients itself and enters the fieldI such molecules are therefore paramagnetic

I Hyperfine coupling constants

AK =⟨0∣∣H(1)

hf

∣∣0⟩ = 8πα2

3

⟨0∣∣δ (rK ) s

∣∣0⟩ ·MK + · · ·

I measure spin density at the nucleusI important in electron paramagnetic resonance (EPR)I recall: there are three hyperfine mechanisms: FC, SD and PSO

I Note: there are no first-order Zeeman or hyperfine couplings for closed-shell molecules

I all expectation values vanish for imaginary operators and triplet operators:⟨c.c.∣∣∣ Ωimaginary

∣∣∣ c.c.⟩≡⟨

c.c.∣∣∣ Ωtriplet

∣∣∣ c.c.⟩≡ 0


Molecular Magnetic Properties Molecular Magnetizabilities

Molecular Magnetizabilities

I Expand the molecular electronic energy in the external magnetic field:

E (B) = E0 − BTM− 1

2BTξB + · · ·

I The magnetizability describes the second-order energy:

ξ = −d2E

dB2= −

⟨0

∣∣∣∣∂2H

∂B2

∣∣∣∣ 0

⟩+ 2

∑n

⟨0∣∣∣ ∂H∂B

∣∣∣ n⟩⟨n ∣∣∣ ∂H∂B

∣∣∣ 0⟩

En − E0

= −1

4

⟨0∣∣∣rTO rOI3 − rOrTO

∣∣∣ 0⟩

︸︷︷︸diamagnetic term

+1

2

∑n

⟨0∣∣LO

∣∣ n⟩ ⟨n ∣∣LTO

∣∣ 0⟩

En − E0︸︷︷︸paramagnetic term

I The magnetizability describes the curvature at zero magnetic field:

linear magnetizability are in fact positive and large enough tomake even the average magnetizability positive !paramag-netic". It is therefore interesting to verify via our finite-fieldLondon-orbital approach whether this very small system isindeed characterized by a particularly large nonlinear mag-netic response. The geometry used for the calculations is thatoptimized at the multiconfigurational SCF level in Ref. 51,corresponding to a bond length of rBH=1.2352 Å.

For the parallel components of the magnetizability andhypermagnetizability, we are able to obtain robust estimatesusing the fitting described above, leading to the values !# =!2.51 a.u. and X# =35.25 a.u., respectively, from aug-cc-pVTZ calculations. The same values are obtained both withLondon orbitals and any common-origin calculation that em-ploys a gauge origin on the line passing through the B and Hatoms since in this case, due to the cylindrical symmetry, theLondon orbitals make no difference.

For the perpendicular components, the estimates of thehypermagnetizability we obtain using the above mentionedfitting procedure are not robust, varying with the number ofdata points included in the least-squares fitting and the de-gree of the polynomial. Using 41 uniformly spaced field val-ues in the range !0.1–0.1 a.u. and a fitting polynomial oforder 16, we arrive at reasonably converged values of !!

=7.1 a.u. and X!=!8"103 a.u. for the magnetizability andhypermagnetizability, respectively, at the aug-cc-pVTZ level.In Fig. 1!c", we report a plot of the aug-cc-pVTZ energy asfunction of field !triangles". For comparison, we report inFig. 1!a" the corresponding benzene plot. As the linear re-sponse for BH is paramagnetic, the curvature of the magneticfield energy dependence is clearly reversed. More impor-tantly, whereas it is evident from Fig. 1!a" that the curve forbenzene is to a very good approximation parabolic so thatthe nonlinearities arise from small corrections that are not

a)

!0.1 !0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

b)

!0.1 !0.05 0 0.05 0.10

2

4

6

8

10

12

14x 10

!3

c)

!0.1 !0.05 0 0.05 0.1

!0.02

!0.018

!0.016

!0.014

!0.012

!0.01

!0.008

!0.006

!0.004

!0.002

0

d)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

!0.03

!0.02

!0.01

0

0.01

0.02

FIG. 1. Energy as a function of the magnetic field for different systems. Triangles represent results from finite-field calculations and solid lines are quarticfitting polynomials. !a" Benzene !with the aug-cc-pVDZ basis" illustrates the typical case of diamagnetic quadratic dependence in response to an out-of-planefield. !b" Cyclobutadiene !aug-cc-pVDZ" deviates from the typical case by exhibiting a nonquadratic dependence on an out-of-plane field. !c" Boronmonohydride !aug-cc-pVTZ" is an interesting case of positive magnetizability for a perpendicular field, exhibiting nonquadratic behavior. !d" Boronmono-hydride !aug-cc-pVTZ" in a larger range of perpendicular fields, exhibiting a clearly nonperturbative behavior.

154114-8 Tellgren, Soncini, and Helgaker J. Chem. Phys. 129, 154114 !2008"

Downloaded 28 Oct 2008 to 129.240.80.34. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

linear magnetizability are in fact positive and large enough tomake even the average magnetizability positive !paramag-netic". It is therefore interesting to verify via our finite-fieldLondon-orbital approach whether this very small system isindeed characterized by a particularly large nonlinear mag-netic response. The geometry used for the calculations is thatoptimized at the multiconfigurational SCF level in Ref. 51,corresponding to a bond length of rBH=1.2352 Å.

For the parallel components of the magnetizability andhypermagnetizability, we are able to obtain robust estimatesusing the fitting described above, leading to the values !# =!2.51 a.u. and X# =35.25 a.u., respectively, from aug-cc-pVTZ calculations. The same values are obtained both withLondon orbitals and any common-origin calculation that em-ploys a gauge origin on the line passing through the B and Hatoms since in this case, due to the cylindrical symmetry, theLondon orbitals make no difference.

For the perpendicular components, the estimates of thehypermagnetizability we obtain using the above mentionedfitting procedure are not robust, varying with the number ofdata points included in the least-squares fitting and the de-gree of the polynomial. Using 41 uniformly spaced field val-ues in the range !0.1–0.1 a.u. and a fitting polynomial oforder 16, we arrive at reasonably converged values of !!

=7.1 a.u. and X!=!8"103 a.u. for the magnetizability andhypermagnetizability, respectively, at the aug-cc-pVTZ level.In Fig. 1!c", we report a plot of the aug-cc-pVTZ energy asfunction of field !triangles". For comparison, we report inFig. 1!a" the corresponding benzene plot. As the linear re-sponse for BH is paramagnetic, the curvature of the magneticfield energy dependence is clearly reversed. More impor-tantly, whereas it is evident from Fig. 1!a" that the curve forbenzene is to a very good approximation parabolic so thatthe nonlinearities arise from small corrections that are not

a)

!0.1 !0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

b)

!0.1 !0.05 0 0.05 0.10

2

4

6

8

10

12

14x 10

!3

c)

!0.1 !0.05 0 0.05 0.1

!0.02

!0.018

!0.016

!0.014

!0.012

!0.01

!0.008

!0.006

!0.004

!0.002

0

d)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

!0.03

!0.02

!0.01

0

0.01

0.02

FIG. 1. Energy as a function of the magnetic field for different systems. Triangles represent results from finite-field calculations and solid lines are quarticfitting polynomials. !a" Benzene !with the aug-cc-pVDZ basis" illustrates the typical case of diamagnetic quadratic dependence in response to an out-of-planefield. !b" Cyclobutadiene !aug-cc-pVDZ" deviates from the typical case by exhibiting a nonquadratic dependence on an out-of-plane field. !c" Boronmonohydride !aug-cc-pVTZ" is an interesting case of positive magnetizability for a perpendicular field, exhibiting nonquadratic behavior. !d" Boronmono-hydride !aug-cc-pVTZ" in a larger range of perpendicular fields, exhibiting a clearly nonperturbative behavior.

154114-8 Tellgren, Soncini, and Helgaker J. Chem. Phys. 129, 154114 !2008"

Downloaded 28 Oct 2008 to 129.240.80.34. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

I left: diamagnetic dependence on the field (ξ < 0); right: paramagnetic dependence on the field (ξ > 0)



Basis-Set Convergence of Hartree–Fock Magnetizabilities

I London orbitals are correct to first-order in the external magnetic field

I For this reason, basis-set convergence is usually improved

I RHF magnetizabilities of benzene:

basis set χxx χyy χzz

London STO-3G −8.1 −8.1 −23.06-31G −8.2 −8.2 −23.1cc-pVDZ −8.1 −8.1 −22.3aug-cc-pVDZ −8.0 −8.0 −22.4

origin CM STO-3G −35.8 −35.8 −48.16-31G −31.6 −31.6 −39.4cc-pVDZ −15.4 −15.4 −26.9aug-cc-pVDZ −9.9 −9.9 −25.2

origin H STO-3G −35.8 −176.3 −116.76-31G −31.6 −144.8 −88.0cc-pVDZ −15.4 −48.0 −41.6aug-cc-pVDZ −9.9 −20.9 −33.9



Mean Absolute Errors for Magnetizabilities

I Mean relative errors (MREs, %) in magnetizabilities of 27 molecules relative to the CCSD(T)/aug-cc-pCV[TQ]Z values.The DFT results are grouped by functional type. The heights of the bars correspond to the largest MRE in each category.(Lutnæs et al., JCP 131, 144104 (2009))



C20 in a Perpendicular Magnetic Field

I All systems become diamagnetic in sufficiently strong fields:

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ææ

ææææææææææææææ

ææ

æææææ

ææææææææææææææ

ææ

æææææææææææ

æææ

ææ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

-0.04 -0.02 0.02 0.04

-756.710

-756.705

-756.700

-756.695

-756.690

-756.685

-756.680


Molecular Magnetic Properties NMR Spin Hamiltonian

High-Resolution NMR Spin Hamiltonian

I Consider a molecule in an external magnetic field B along the z axis and with nuclearspins IK related to the nuclear magnetic moments MK as:

MK = γK~IK ≈ 10−4 a.u.

where γK is the magnetogyric ratio of the nucleus.

I Assuming free molecular rotation, the nuclear magnetic energy levels can be reproduced bythe following high-resolution NMR spin Hamiltonian:

HNMR = −∑K

γK~(1− σK )BIK z︸︷︷︸nuclear Zeeman interaction

+∑K>L

γKγL~2KKLIK · IL︸︷︷︸nuclear spin–spin interaction

where we have introduced

I the nuclear shielding constants σKI the (reduced) indirect nuclear spin–spin coupling constants KKL

I This is an effective nuclear spin Hamiltonian:

I it reproduces NMR spectra without considering the electrons explicitlyI the spin parameters σK and KKL are adjusted to fit the observed spectraI we shall consider their evaluation from molecular electronic-structure theory


Molecular Magnetic Properties NMR Spin Hamiltonian

Simulated 200 MHz NMR spectra of Vinyllithium 12C2H36Li


Molecular Magnetic Properties Nuclear Shielding Constants

Nuclear Shielding Constants

I Expansion of closed-shell energy in an external field B and nuclear magnetic moments MK :

E (B,M) = E0 +1

2BTE(20)B +

1

2

∑K

BTE(11)K MK +

1

2

∑KL

MTK E

(02)KL ML + · · ·

I Here E(11)K describes the coupling between the applied field and the nuclear moments:

I in the absence of electrons (i.e., in vacuum), this coupling is identical to −I3:

Hnucz = −B ·

∑K

MK ← the purely nuclear Zeeman interaction

I in the presence of electrons (i.e., in a molecule), the coupling is modified slightly:

E(11)K = −I3 + σK ← the nuclear shielding tensor

I Shielding constants arise from a hyperfine interaction between the electrons and the nuclei

I they are of the order of α2 ≈ 5 · 10−5 and are measured in ppm

I The nuclear Zeeman interaction does not enter the electronic problem

I compare with the nuclear–nuclear Coulomb repulsion




PSO hyperfine

Zeeman

PSOFC+SDFC+SD

SS, SO, OO

SO SO



Ramsey’s Expression for Nuclear Shielding Tensors

I Ramsey’s expression for nuclear shielding tensors of a closed-shell system:

σK =d2Eel

dBdMK=

⟨0

∣∣∣∣ ∂2H

∂B∂MK

∣∣∣∣ 0

⟩− 2

∑n

⟨0∣∣∣ ∂H∂B

∣∣∣ n⟩⟨n ∣∣∣ ∂H∂MK

∣∣∣ 0⟩

En − E0

=α2

2

⟨0

∣∣∣∣∣ rTO rK I3 − rOrTKr3K

∣∣∣∣∣ 0

⟩︸︷︷︸

diamagnetic term

−α2∑n

⟨0∣∣LO

∣∣ n⟩ ⟨n ∣∣∣r−3K LT

K

∣∣∣ 0⟩

En − E0︸︷︷︸paramagnetic term

I The (usually) dominant diamagnetic term arises from differentiation of the operator:

AO · AK =1

2α2r−3

K (B× rO) · (MK × rK )

I As for the magnetizability, there is no spin contribution for singlet states:

S |0〉 ≡ 0 ← singlet state

I For 1S systems (closed-shell atoms), the paramagnetic term vanishes completely and theshielding is given by (assuming gauge origin at the nucleus):

σLamb =1

3α2⟨

1S∣∣∣r−1K

∣∣∣ 1S⟩← Lamb formula



Benchmark Calculations of BH Shieldings (ppm)

σ(11B) ∆σ(11B) σ(1H) ∆σ(1H)HF −261.3 690.1 24.21 14.15MP2 −220.7 629.9 24.12 14.24CCSD −166.6 549.4 24.74 13.53CCSD(T) −171.5 555.2 24.62 13.69CCSDT −171.8 557.3 24.59 13.72CCSDTQ −170.1 554.7 24.60 13.70CISD −182.4 572.9 24.49 13.87CISDT −191.7 587.0 24.35 14.06CISDTQ −170.2 554.9 24.60 13.70FCI −170.1 554.7 24.60 13.70

I TZP+ basis, RBH = 123.24 pm, all electrons correlated

I J. Gauss and K. Ruud, Int. J. Quantum Chem. S29 (1995) 437

I M. Kallay and J. Gauss, J. Chem. Phys. 120 (2004) 6841



Coupled-Cluster Convergence of Shielding Constants in CO (ppm)

CCSD CCSD(T) CCSDT CCSDTQ CCSDTQ5 FCI

σ(13C) 32.23 35.91 35.66 36.10 36.14 36.15∆σ(13C) 361.30 356.10 356.47 355.85 355.80 355.79σ(17O) −13.93 −13.03 −13.16 −12.81 −12.91 −12.91

∆σ(17O) 636.01 634.55 634.75 634.22 634.52 634.35

I All calculations in the cc-pVDZ basis and with a frozen core.

I Kallay and Gauss, J. Chem. Phys. 120 (2004) 6841.



Calculated and Experimental Shielding Constants (ppm)

HF CAS MP2 CCSD CCSD(T) exp.HF F 413.6 419.6 424.2 418.1 418.6 410± 6 (300K)

H 28.4 28.5 28.9 29.1 29.2 28.5± 0.2 (300K)H2O O 328.1 335.3 346.1 336.9 337.9 323.6± 6 (300K)

H 30.7 30.2 30.7 30.9 30.9 30.05± 0.02NH3 N 262.3 269.6 276.5 269.7 270.7 264.5

H 31.7 31.0 31.4 31.6 31.6 31.2± 1.0CH4 C 194.8 200.4 201.0 198.7 198.9 198.7

H 31.7 31.2 31.4 31.5 31.6 30.61F2 F −167.9 −136.6 −170.0 −171.1 −186.5 −192.8N2 N −112.4 −53.0 −41.6 −63.9 −58.1 −61.6± 0.2 (300K)CO C −25.5 8.2 10.6 0.8 5.6 3.0± 0.9 (eq)

O −87.7 −38.9 −46.5 −56.0 −52.9 −56.8± 6 (eq)

I For references and details, see Chem. Rev. 99 (1999) 293.

I for exp. CO and H2O values, see Wasylishen and Bryce, JCP 117 (2002) 10061



NMR: Mean Absolute Errors Relative to Experiment

I Mean absolute errors relative to experimental (blue) and empirical equilibrium values (red)

MAE (Exp.)

MAE (Emp. Eq.)

0

5

10

15

20

25

30

MA

E /

pp

m

MAE (Exp.) MAE (Emp. Eq.)

I Kohn–Sham calculations give shielding constants of uneven qualityI errors increase when vibrational corrections are appliedI Teale et al. JCP 138, 024111 (2013)


Molecular Magnetic Properties Indirect Nuclear Spin–Spin Coupling Constants

Direct and Indirect Nuclear Spin–Spin Couplings

I The last term in the expansion of the molecular electronic energy in B and MK

E (B,M) = E0 + 12

BTE(20)B + 12

∑K BTE

(11)K MK + 1

2

∑KL MT

K E(02)KL ML + · · ·

describes the coupling of the nuclear magnetic moments in the presence of electrons

I There are two distinct contributions to the coupling:the direct and indirect contributions

E(02)KL = DKL + KKL

I The direct coupling occurs by a classical dipole mechanism:

DKL = α2R−5KL

(R2KLI3 − 3RKLRT

KL

)∼ 10−12 a.u.

I it is anisotropic and vanishes in isotropic media such as gases and liquids

I The indirect coupling arises from hyperfine interactions with the surrounding electrons:

– it is exceedingly small: KKL ∼ 10−16 a.u. ∼ 1 Hz– it does not vanish in isotropic media– it gives the fine structure of high-resolution NMR spectra

I Experimentalists usually work in terms of the (nonreduced) spin–spin couplings

JKL = h γK2π

γL2π

KKL ← isotope dependent




PSO hyperfine

Zeeman

PSOFC+SDFC+SD

SS, SO, OO

SO SO



Ramsey’s Expression for Indirect Nuclear Spin–Spin Coupling Tensors

I The indirect nuclear spin–spin coupling tensors of a closed-shell system are given by:

KKL =d2Eel

dMKdML=

⟨0

∣∣∣∣ ∂2H

∂MK∂ML

∣∣∣∣ 0

⟩− 2

∑n

⟨0∣∣∣ ∂H∂MK

∣∣∣ n⟩⟨n ∣∣∣ ∂H∂ML

∣∣∣ 0⟩

En − E0

I Carrying out the differentiation of the Hamiltonian, we obtain Ramsey’s expression:

KKL = α4

⟨0

∣∣∣∣∣ rTK rLI3 − rK rTLr3K r

3L

∣∣∣∣∣ 0

⟩︸︷︷︸

diamagnetic spin–orbit (DSO)

− 2α4∑n

⟨0∣∣∣r−3K LK

∣∣∣ n⟩⟨n ∣∣∣r−3L LT

L

∣∣∣ 0⟩

En − E0︸︷︷︸paramagnetic spin–orbit (PSO)

− 2α4∑n

⟨0

∣∣∣∣ 8π3δ(rK )s +

3rK rTK−r2K I3

r5K

s

∣∣∣∣ n⟩⟨n ∣∣∣∣ 8π3δ(rL)sT+

3rLrTL −r2L I3

r5L

sT∣∣∣∣ 0

⟩En − E0︸︷︷︸

Fermi contact (FC) and spin–dipole (SD)

I the isotropic FC/FC term often dominates short-range coupling constantsI the FC/SD and SD/FC terms often dominate the anisotropic part of KKLI the orbital contributions (especially DSO) are usually but not invariably smallI for large internuclear separations, the DSO and PSO contributions cancel



Relative Importance of Contributions to Spin–Spin Coupling Constants

I The isotropic indirect spin–spin coupling constants can be uniquely decomposed as:

JKL = JDSOKL + JPSO

KL + JFCKL + JSD

KL

I The spin–spin coupling constants are often dominated by the FC term

I Since the FC term is relatively easy to calculate, it is tempting to ignore the other terms.

I However, none of the contributions can be a priori neglected (N2 and CO)!

H2 HF H2O

O-H

NH3

N-H

CH4

C-H

C2H4

C-C

HCN

N-C

N2 CO C2H2

C-C

-100

0

100

200

FC

FCFC FC FC FC

FCFC

FCFC

PSO

PSO

PSO

SD

SD

SD



Calculation of Indirect Nuclear Spin–Spin Coupling Constants

I The calculation of spin–spin coupling constants is a challenging task

I Spin–spin coupling constants depend on many coupling mechanisms:

I 3 singlet response equations and 7 triplet equations for each nucleusI for shieldings, only 3 equations are required, for molecules of all sizes

I Spin–spin coupling constants require a proper description of static correlation

I the Hartree–Fock model fails abysmallyI MCSCF theory treats static correlation properly but is expensive

I Spin–spin couplings are sensitive to the basis set

I the FC contribution requires an accurate electron density at the nucleiI steep s functions must be included in the basis

I Spin–spin couplings are sensitive to the molecular geometry

I equilibrium structures must be chosen carefullyI large vibrational corrections (often 5%–10%)

I For heavy elements, a relativistic treatment may be necessary.

I However, there is no need for London orbitals since no external magnetic field is involved.



Restricted Hartree–Fock Theory and Triplet Instabilities

I The correct description of triplet excitations is important for spin–spin coupling constants

I In restricted Hartree–Fock (RHF) theory, triplet excitations are often poorly describedI upon H2 dissociation, RHF does not describe the singlet ground state correctlyI but the lowest triplet state dissociates correctly, leading to triplet instabilities

2 4 6 R

-2

-1

1

1Sg+H1Σg

2L

1Sg+H1Σu

2L

1Sg+HFCIL

1Sg+HFCIL

3Su+H1Σg1ΣuL

1Su+H1Σg1ΣuL

covalent

ionic

cov-ion

1SH1s2L

3PH1s2pL

1PH1s2pL

1DH2p2L

I Near such instabilities, the RHF description of spin interactions becomes unphysical

C2H4/Hz 1JCC1JCH

2JCH2JHH

3Jcis3Jtrans

exp. 68 156 −2 2 12 19RHF 1270 755 −572 −344 360 400CAS 76 156 −6 −2 12 18B3LYP 75 165 −1 3 14 21



Reduced Spin–Spin Coupling Constants by Wave-Function Theory

RHF CAS RAS SOPPA CCSD CC3 exp∗ vib

HF 1KHF 59.2 48.0 48.1 46.8 46.1 46.1 47.6 −3.4CO 1KCO 13.4 −28.1 −39.3 −45.4 −38.3 −37.3 −38.3 −1.7N2

1KNN 175.0 −5.7 −9.1 −23.9 −20.4 −20.4 −19.3 −1.1H2O 1KOH 63.7 51.5 47.1 49.5 48.4 48.2 52.8 −3.3

2KHH −1.9 −0.8 −0.6 −0.7 −0.6 −0.6 −0.7 0.1NH3

1KNH 61.4 48.7 50.2 51.0 48.1 50.8 −0.32KHH −1.9 −0.8 −0.9 −0.9 −1.0 −0.9 0.1

C2H41KCC 1672.0 99.6 90.5 92.5 92.3 87.8 1.21KCH 249.7 51.5 50.2 52.0 50.7 50.0 1.72KCH −189.3 −1.9 −0.5 −1.0 −1.0 −0.4 −0.42KHH −28.7 −0.2 0.1 0.1 0.0 0.2 0.03Kcis 30.0 1.0 1.0 1.0 1.0 0.9 0.13Ktns 33.3 1.5 1.5 1.5 1.5 1.4 0.2∣∣∆∣∣ abs. 180.3 3.3 1.6 1.8 1.2 1.6 ∗at Re

% 5709 60 14 24 23 6

I SOPPA: second-order polarization-propagator approximation



Comparison of Density-Functional and Wave-Function Theory

I Normal distributions of errors for indirect nuclear spin–spin coupling constantsI for the same molecules as on the previous slides

-30 30

CAS

-30 30 -30 30

RAS

-30 30 -30 30

SOPPA

-30 30 -30 30

CCSD

-30 30

-30 30

HF

-30 30

LDA

-30 30 -30 30

BLYP

-30 30 -30 30

B3LYP

-30 30

I Some observations:I LDA underestimates only slightly, but has a large standard deviationI BLYP reduces the LDA errors by a factor of twoI B3LYP errors are similar to those of CASSCFI The CCSD method is slightly better than the SOPPA method


Documents

Derivatives and Properties - Universitetet i oslofolk.uio.no/helgaker/talks/MWM16_DerivativesProperties.pdf · Trygve Helgaker (CTCC, University of Oslo) Derivatives and Properties