Advanced Electronic Structure Theory

David P. Tew!

Warwick 2016

Overview

These lectures introduce theoretical methods designed for computing electronic energies of molecular systems. You will learn about the challenges that electron correlation presents and the extent to which state-of-the-art techniques meet these challenges.!1.1Exact conditions1.2Hartree-Fock Theory1.3Configuration Interaction Theory

2.1Coupled-cluster Theory2.2Explicit electron correlation2.3Multi-reference Theory!

Primary Goal

We aim to solve the electronic Schrödinger equation for as many of the particles involved in the processes under investigation as necessary/possible, to as high accuracy as necessary/possible!!!!!

!

This is a prerequisite for studying the thermodynamics and kinetics of the system (e.g. Transition State Theory or Direct Dynamics) and is often the starting point for more accurate treatments that incorporate relativistic and non-adiabatic effects.

H = �X

i

1

2r2

i �X

iI

Zj

|ri �RI |+X

i>j

1

|ri � rj |

H = E

~ = 4⇡✏0 = me = e = 1

(r1, s1, . . . , rn, sn)

An exact solution : the hydrogen atom

The Schrödinger equation for the H atom can be solved exactly using spherical polar coordinates.!The solutions are the atomic orbitals and have quantum numbers n,l,ml,ms which gives rise to the shell structure of excited states of hydrogen. The electronic wave function is the product of a spatial part (the AO) and a spin part for spin up or spin down electrons

1s2s 2p3s 3p 3d4s 4p 4d 4f…

�(r, s) = '(r)�(s)

0

-0.25

-0.50

Exact conditions 1

The exact wave function is also an eigenfunction of any operator that commutes with the Hamiltonian!!This includes:

• Spin is a good quantum number• • Molecular point group symmetry is obeyed• Fermionic antisymmetry is satisfied

!Note: is independent of spin and is invariant to permutations among electron coordinates and among nuclear coordinates

If H = E and [A, H] = 0, then A = a

S2 S2 = S(S + 1)

Sz = Ms Sz

Cn, �v . . .

⇡µ

H

H = �X

i

1

2r2

i �X

iI

Zj

|ri �RI |+X

i>j

1

|ri � rj |

Exact conditions 2

The local energy is constant!Aside: This still holds at points infinitesimally close to nodal planes where

!The Coulomb energy diverges when an electron and nucleus meet!!

The kinetic energy must exactly cancel this singularity. This imposes a universal nuclear cusp condition:

@

@r

����r=0

= �Z r=0

= r=0(1� Zr + . . . ) 0

0.1

0.2

0.3

0.4

0.5

0.6

-4 -3 -2 -1 0 1 2 3 4

q 1s(

z 1)

z1

✓�1

2r2

1 �Z1

|R1 � r1|+ O

◆ (r,�)|r1=R1 = E (r,�)|r1=R1

Eloc

(r,�) ⌘ H (r,�)

(r,�)= E

(r,�) = 0

Exact conditions 3

Consider the regions where one electron is far away from the molecular framework and the rest are close. The wave function is then an antisymmetrised product of that of the cation and a distant electron!!The local energy is the sum of the energies of the cation and the distant electron!!The long-range behaviour of is therefore that of a hydrogen-like 1s orbital with effective charge

(x) = A[ e(x1) M+

(x2 . . .xn)]

E = Ee+ EM+

so Ee= EIP

p2EIP

e(x)

e(x) ! e�p2EIP r

The variational principle and trial wave functions

According to the Rayleigh-Ritz variational principal the energy of a trial wave function is never below the exact energy.!!!In a numerical approximation the unknown n-electron exact wave function is expanded as a linear combination of known n-electron functions (n-electron basis functions).!!!The expansion has linear expansion coefficients and the basis functions contain non-linear parameters . Either or both can be optimised by minimising the energy to obtain the best possible trial wave function using that set of functions.

Eexact

h |H| ih | i

=X

P

CP P

CP

P (r1, . . . , rn,�1, . . . ,�n; �)

�

Molecular Orbital Theory (Hartree-Fock)

By assuming that the electrons are independent, we approximate the n-electron wavefunction as a product of molecular orbitals.

The valid n-electron wave function satisfying Fermi antisymmetry is an antisymmetrised product of one-electron spin-orbitals!!!!!!!!The theory that optimises the orbitals to minimise the energy is Hartree-Fock theory.

=1pn!

���������

�1(1) �2(1) · · · �n(1)�1(2) �2(2) · · · �n(2)

......

. . ....

�1(n) �2(n) · · · �n(n)

���������

=1pn!A(�1(1)�2(2) · · ·�n(n))

= |�1�2 · · ·�ni

The HF energy

The HF orbitals are orthonormal so that The energy is therefore!!It contains a one-electron term and a two-electron term!!!Using the Slater-Condon rules, the energy expression simplifies into integrals involving at most two orbitals at a time.

h�i|�ji = �ij h | i = 1

hi = �1

2r2

i �X

I

ZI

|ri �RI |

E = h |H| i = h |X

i

hi| i+ h |X

i<j

Vij | i

Vij =1

|ri � rj |=

1

rij

E =X

i

h�i|h|�ii+1

2

X

ij

h�i�j |V |�i�j � �j�ii

The HF equations

Minimising the energy with respect to orthonormal orbitals leads to a set of coupled orbital equations, known as the Fock equations!where !!!!!Each electron experiences an average potential from the others and the equations must be solved iteratively until self-consistency!The eigenvalues are the energies of the orbitals (MO theory)

(h+ J � K)�i = ✏i�i

J =X

j

Zd2�j(2)r

�112 �j(2)

K =X

j

Zd2�j(2)r

�112 ⇡12�j(2)

Coulomb operator

!Exchange operator

✏i

MO theory

The HF orbitals and energies provide a qualitative description of bonding in molecules. E.g. water !

Notation: refers to an occupied MO refers to an unoccupied MO

Koopmans theorem:The energy differences!!are a good approximationto the excitation energies

✏a � ✏i

�i

�a

Gaussian-type orbital functions

The orbitals in HF theory are efficiently expressed as a linear combination of atom-centred Gaussian-type orbitals !!

The index u labels the centre I, the exponent n and the angular function l,m!

!!!!!!

• Gaussian rather than Slater orbitals are chosen for computational convenience

• Atom-centred functions directly fit the nuclear cusp and can mimic polarised hydrogenic shell structure

• The expansion is formally complete (in fact over-complete if multiple atomic centres are involved)

�Inlm(r�RI) = e�↵nr

2

Ylm(✓,�)�(r) =X

µ

Cµ �µ(r)

Gaussian-type orbitals reproducing the hydrogen 1s orbital

Basis sets

We are in the very fortunate position that a number of people have spent an enormous amount of time and energy creating reliable basis sets for every element of the periodic table

SCF basis sets with Cardinal number X have X functions per occupied orbital e.g. for the C atomDZ : 4s2p TZ : 5s3p QZ : 6s4p

To account for a non-spherical (molecular) environment, polarisation functions are addedDZ : 1d TZ : 2d1f QZ : 3d2f1gTo account for diffuse electron clouds (halogens and anions), the basis is augmented with 1 diffuse function per angular momentum

For post-HF calculations, correlation-consistent basis sets are recommended since they are an optimal compromise between HF and correlation requirements.

Convergence for one-electron systems and the IPM

The GTO expansion is rapidly converging.*

The error in the energy of one-electron systems decays exponentially with the number N of Gaussian functions,* and thus exponentially with the Cardinal number X of the basis set:!!The error can thus be reduced through extrapolation!!

This exponential convergence is shared by all independent particle models (HF and DFT)* if the exponents obey a well-tempered sequence

* Be careful! Use polarisation and diffuse functions when needed

E(X) = Eexact

+A exp(�BX)

E1 =E2

Y � EXEZ

2EY � EX � EZY = X � 1 Z = X � 2

Electron correlation

In HF theory each electron responds only to the average potential of the other electrons (an independent particle model).

The wavefunction does not account for electron correlation arising from the instantaneous Coulombic repulsions.!• Too small dissociation energies (F2 is not even bound)

• Much too high reaction barriers

• Dispersion effects completely missing

• Bond lengths 2 pm too short

• Frequencies 10% too large

• Dipole moments 10% too large

!

HF theory is often qualitatively correct, but inaccurate

Configuration Interaction Theory

The unknown n-electron exact wave function is expanded as a linear combination of known n-electron basis functions : SDs with all possible arrangements of the electrons the HF orbitals!!

The Secular Equations that result from applying the variational principal to the CI wave function are particularly simple

!

!!

The eigenvalues of H are the ground and excited electronic states

The matrix elements are 2e-integrals, just like in HF theory

=X

P

CP P P = |�P1�P2 . . .�Pni

X

Q

HPQCQ = EX

Q

SPQCQ

SPQ = �PQ

HPQ = 0 if P and Q differ by more than 2 orbitals

HPQ

Slater Determinants

Slater Determinants are a very convenient choice for the n-electron basis functions.!!

The index P labels which spin-orbitals are occupied in each SD function!Each n-elec. basis function individually obeys the exact conditions

• Is a pure spin state with definite S and Ms*

• Obeys the molecular point group symmetry

• Has full antisymmetric permutational symmetry!

Moreover, an expansion in SDs is complete in the limit of a complete set of orbitals (the error can be made arbitrary small by increasing the number of functions in the expansion)

* Some spin states require specific combinations of SDs, called CSFs.

=X

P

CP P P = |�P1�P2 . . .�Pni

CI wavefunction for helium using 1s, 2s, 2p orbitals

The contribution from the 2s orbitals decreases the probability of the electrons being at the same radius : radial correlationThe contribution from the 2p orbitals decreases the probability of the electrons being on the same side : angular correlation

-2 -1 0 1 2 -2-1

01

20.00.10.20.30.40.5

-2 -1 0 1 2 -2-1

01

2-0.020

-0.010

0.000

0.010

-2 -1 0 1 2 -2-1

01

2-0.010-0.0050.0000.0050.010

-2 -1 0 1 2 -2-1

01

2-0.020

-0.010

0.000

0.010

x1

y1e2

e2

e2e2

x1

x1

x1

y1 y1

y1

(a) CI (1s2s2p) (b) 2s correlation

(c) 2p correlation (d) 2s+2p correlation

The cost of CI

Consider the example of the H2O molecule in a TZVPP basis. There are 59 AOs, and therefore 59 MOs (5 occupied, 54 virtual).

Each configuration (10 electrons in 118 spin-orbitals) can be represented through an occupation number vector

!!!!!!

The total number of configurations is !Just storing one CI vector requires 200 Tb of memory!

|1, 1, 1, 1, 1, 0, 0, 0, 0, · · · , 1, 1, 1, 1, 1, 0, 0, 0, 0, · · · i|1, 1, 1, 1, 0, 1, 0, 0, 0, · · · , 1, 1, 1, 1, 1, 0, 0, 0, 0, · · · i|1, 1, 1, 1, 0, 1, 0, 0, 0, · · · , 1, 1, 1, 1, 0, 0, 1, 0, 0, · · · i|0, 1, 0, 1, 0, 1, 1, 0, 1, · · · , 1, 0, 1, 1, 0, 0, 1, 1, 0, · · · i

HF

1 excitation

2 excitations

5 excitations

�1,�2,�3,�4,�5,�6,�7,�8,�9, · · · �1,�2,�3,�4,�5,�6,�7,�8,�9, · · ·↵ �

✓59

5

◆⇥✓59

5

◆= 2.5⇥ 1013

Truncated CI

Assuming that the HF wavefunction is a good starting point, we can rank the configurations by their degree of difference from the HF reference . Not all configurations are equally important.

Formally, this is revealed through a Krylov subspace expansion!!!!Since the Hamiltonian is a 2-body operator, populates configurations with double excitations from HF. These describe pair correlations and are the most important configurations.A further application of generates singles, triples and quadruples, describing orbital relaxation, 3- and 4-electron correlations, etc.

Km = span{C0, HC0, H2C0, . . . , H

m�1C0}

HC0

C0

H

= =

CISD

Truncating the full configuration space to only include single and double excitations results in the CISD method. This wavefunction can describe pair correlations and orbital relaxations.

The CISD wavefunction is!

!

This is obviously extended to give CISDT, CISDTQ…

The same wavefunction in the language of second quantisation is

!!Where the excitation operators generate a singly and doubly excited configuration from the HF reference

|CISDi =⇣C0 +

X

ia

Ciaa

†aai +

X

i<j,a<b

Cijaba

†aa

†baiaj

⌘|HFi

|CISDi = C0|HFi+X

ia

Cia|aii+

X

i<j,a<b

Cijab|abiji

|aii = a†aai|HFi |abiji = a†aa†baiaj |HFi

Coupled Cluster theory

In truncated CI theory, the wave function is expanded as e.g.!!

In coupled cluster theory, the CI coefficients of the n-body excitations are generated as products of few-body excitations!!!!!!Since the CC wave function contains excitations in all orbitals simultaneously, a variational treatment has the same cost as FCI.

Three-body terms are generated from T1T2 +

16 T

31

|CISDi = C0|HFi+X

ia

Cia|aii+

X

i<j,a<b

Cijab|abiji

|CCSDi = eT1+T2 |HFi

T1 =X

ia

tiaa†aai

T2 =X

i<j,a<b

tijaba†aa

†baiaj

Coupled Cluster theory in first quantisation

To better understand coupled cluster theory it is instructive to examine the CCD wave function for an example: helium dimer!!!!!!

• Every orbital pair has a correlation function

• The wave function contains all possible products of orbitals and correlation functions

• The red terms are those missing from the CID wave function and ensure size extensivity:

|HFi = |1sA↵1sA� 1sB↵ 1sB� i HeA … HeB

|CCDi = |1sA↵1sA� 1sB↵ 1sB� i+ |✓AA

↵� 1sB↵ 1sB� i � |✓AB

↵↵ 1sA� 1sB� i+ |✓AB

↵� 1sA� 1sB↵ i

+ |1sA↵✓AB�↵ 1sB� i � |1sA↵✓AB

�� 1sB↵ i+ |1sA↵1sA� ✓BB↵� i

+ |✓AA↵� ✓BB

↵� i � |✓AB↵↵ ✓AB

�� i+ |✓AB↵� ✓AB

�↵ i�i�j

|CCDi R=1�! |CCDAi|CCDBi

✓ij =X

a<b

tijab|abi

The CC working equations

The energy is obtained through projecting the SE with !!!

This energy is not variational for truncated CC

The amplitudes are obtained through projection with !!!

is the similarity transformed Hamiltonian, which is!

The amplitude equations are coupled 4th order polynomials

hHF|

H|CCi ⇡ E|CCi ! ECC = hHF|H|CCi

hai |e�T HeT |HFi = 0

habij |e�T HeT |HFi = 0

hai |e�T

e�T HeT

ECC = EHF +X

ijab

(tijab + tiatjb)Liajb

H + [H, T ] + 12 [[H, T ], T ] + 1

6 [[[H, T ], T ], T ] + 124 [[[[H, T ], T ], T ], T ]

singles

doubles

Convergence of the CI and CC hierarchies

!!!!!!!!The CI and CC series provide a hierarchy of methods with increasing level of n-body correlation that converge to FCI, where all correlation effects are included (within a finite basis set).

The convergence is exponential if the IMP is a good starting point.

* plot of error (Eh) w.r.t. FCI for H2O using cc-pVDZ basis

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

S D T Q 5 6

Erro

r

Truncation

CICC Excitation CI CC

1,2-body CISD CCSD

3-body CISDT CCSDT

4-body CISDTQ CCSDTQ

5-body CISDTQ5 CCSDTQ5

=X

P

CP P = eT 0

Comparison of basis set error and truncation error

!!!!!!!!The basis set incompleteness error in the 2-body energy is larger than the error incurred by neglecting 3- and higher-body effects. At least a cc-pV5Z basis is required to obtain accurate energies. (cc-pVQZ for energy differences since is basis set errors cancel to a larger extent than truncation errors).

* plot of error (Eh) w.r.t. CCSD basis set limit for H2O

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

S D T Q 5 6

Erro

r

Truncation

CICC

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

DZ TZ QZ 5Z 6Z

Erro

r

X

cc-pVXZ

Comparison of basis set error and truncation error

Error statistics (kJ/mol) w.r.t experiment for 14 reaction energies of small molecules involving main group elements.!!!!!!!!!

(T) refers to a perturbative approximation to the triples energy

Large basis CCSD(T) calculations return accuracy of ~5 kJ/mol

Comparison to post-FCI contributions

An illustrative example The electron affinity of oxygen 1.4611 eV

!!!!!!!!!

!CCSD basis set incompleteness is larger than post CCSD(T) terms

• Mass-Velocity-Darwin (MVD)

• Spin-orbit coupling

• Electron-nucleus dynamic coupling (non-adiabatic effects)

– 0.0024

– 0.0059

+ 0.0001

= 1.4610 eV

CCSD T Q P EA ΔCCSD EA

1.2654 0.1727 0.0121 -0.0008 1.4495 0.0197 1.4692

CCSDT - CCSD (5Z)CCSDTQ - CCSDT (TZ)CCSDTQP - CCSDTQ (DZ)CCSD/CBS - CCSD (5Z)

T =Q = P =

CCSD = �

��

�

Electron correlation — lessons from helium

Although the SD expansion for two electron systems is formally complete, it is very slowly converging, even for helium!!!!!!!!!

Extrapolation :

error ∝ N–1

∝ X–3

P = |�P1�P2i =X

P

CP P

0.0001

0.001

0.01

0.1

0 50 100 150 200 250 300 350 400 450 500

Erro

r (E

h)

Number of basis functions

X IP (eV)

D 24.152

T 24.496

Q 24.555

limit 24.591

exp. 24.587

Y = X � 1

E1 =X3EX � Y 3EY

X3 � Y 3Y = X � 1

Electron correlation — lessons from helium

!!!!!!!!!!Consider a cut of the wave function where both electrons are at a radius of 0.5 a0. The exact wave function has a cusp when the two electrons coalesce. The smooth orbitals used in the SD expansion are unable to represent this sharp correlation cusp efficiently.

θ12

r = 0.5 a1 0

r1

r2

He2+

CI angular correlation

0.20

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.30

π0.5π0-0.5π-π

\ HF

exact

CI nmax

34

65

7T12

R12/F12 theory

Kutzelnigg’s observations:1. Only the short-range part of the pair correlation function is

problematic—the standard expansion is sufficient for the medium and long-range correlation.

2. The 3-, 4- and 5-electron integrals that arise can be approximated to sufficient accuracy using only 2-electron integrals by employing a RI technique.

!In F12 theory the pair function is expanded as*!!The standard orbital expansion is augmented with a single fixed term that explicitly satisfies the cusp ( is used for triplet pairs).

* In the original R12 theory

|✓iji =X

ab

tijab|�a�bi+ 12 Q12f(r12)|�i�ji

14

f(r12) = r12

The correlation factor

For ortho and para helium it is possible to compute the optimum correlation factor in

!

!

!

!

!The function reproduces the short- and medium-range correlation hole very closely. is a length-scale parameter.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5

f(r12

)

r12 (a0)

fopt1/2a(1-exp-ar12)

0.5 r12 0

0.1

0.2

0.3

0 1 2 3 4 5f(r

12)

r12 (a0)

fopt1/4a(1-exp-ar12)

0.25 r12

1S 3SValance 0.8-1.4

Core-Val. 1.8-2.6

Core 1.8-2.6

�

-0.5 0.0 0.5 1.0 1.5 -1.0-0.5

0.00.51.0

-0.06-0.04-0.020.000.02

-0.5 0.0 0.5 1.0 1.5 -1.0-0.5

0.00.51.0

-0.06-0.04-0.020.000.02

-0.5 0.0 0.5 1.0 1.5 -1.0-0.5

0.00.51.0

-0.06-0.04-0.020.000.02

�

1� (1� e��r12)

✓ 12f(r12)1s1s

|✓iji =X

ab

tijab|�a�bi+ 12 Q12f(r12)S|�i�ji

f(r12)

F12 Theory

F12 doubles appear alongside standard doubles in the equations!!!!!!

There are no additional amplitude equations, but the (non-zero) F12 residual is added as an energy correction (~Lagrangian)

⌧ai |HFi = |ai i⌧abij |HFi = |abij i

12

X

�

Rij�⌧

�ij |HFi = |f12ij i

|CCSD-F12i = eT1+T2+T20 |HFi T1 =X

ia

tia⌧ai

T2 = 14

X

ijab

tijab⌧abij

T20 =14

X

ij�

Rij�⌧

�ij

ECC-F12 = hHF|H|CC-F12i+ 12

X

ij

hf12ij |e�T HeT |HFi

hai |e�T HeT |HFi = 0

habij |e�T HeT |HFi = 0

singles

doubles

Performance

!!!!!!!!The basis set incompleteness error in the 2-body energy is now smaller than the error incurred by neglecting 3- and higher-body effects, even when small (cc-pVDZ-F12) basis sets are used.!CCSD(T)-F12 is now a standard method in quantum chemistryTURBOMOLE, MOLPRO, DALTON, ORCA, GELLAN

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

DZ TZ QZ 5Z 6Z

Erro

r

X

CCSD/cc-pVXZCCSD(F12*)/cc-pVXZ-F12

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

S D T Q 5 6

Erro

r

Truncation

CICC

Other explicitly correlated methods

Strategies for explicit r12 dependence in pair functions

• Use functions with simple integral formulae and introduce a high degree of flexibility

• Use physically motivated basis functions and rely on mathematical ingenuity for the many-electron integration!

! Method Scaling Accuracy Max. Size

Hyl-CI n! nEh 5 electrons

EGC n! nEh 4 electrons

VMC n3 85% Ecorr 500 atoms

DMC n3 90-98% Ecorr 500 atoms

CCSD-GTG n7 uEh 5 atoms

CCSD(T)/5Z n7 99.8% Ecorr 10 atoms

CCSD(T)-F12/DZ n7 99.8% Ecorr 30 atoms

Strong correlation

The CCSD(T) method works well when the HF reference is a good starting point.

Consider the example of the H2 molecule in the basis of 1s orbitals

!

!

!

!

The HF reference is qualitatively wrong when there are two or more interacting states close in energy : strong correlation

prevalent in stretched bonds, transition metals, radicals

-1.5

-1

-0.5

0

0.5

0 1 2 3 4 5 6 7 8

Ener

gy (

E h)

Bond Length (a0)

(b)

|HFi =|�g↵�g�i

|CIi =c0|�g↵�g�i+ c1|�u↵�u�i

HFCI

CASSCF

The generalisation of the HF method to the situation where more than one SD is required for a qualitatively correct description of the electronic structure is called complete active space self-consistent-field (CASSCF). !!!!!!!A CI calculation is performed in the configurations of n active electrons in m active orbitals, with an associated optimisation of all of the orbitals involved. Scales exponentially with the size of the active space, but cubically with the size of the system.

active

inactive

virtual

fermi level

|CASSCFi = A|CIi|IPMi

|CIi

|IPMi

DMRG and FCI-QMC

The CASSCF method is very appealing, but the pragmatic decisions imposed by the exponential cost of increasing the active space often makes it very difficult to define consistent orbital spaces across reaction pathways, particularly if several electronic states are traversed.

In Density Matrix Renormalisation Group (DMRG) the CI vector is approximated as a tensor train with one tensor per orbital, reducing cost of storage and manipulation

!

!

In FCI-QMC, the CI vector is sampled by a Monte-Carlo dynamic where energies are extracted through time averaging

| i =X

P

CP |P i CP 'X

l1,l2,...,lm

C1l1C

2l2 . . . C

mlm

| i ' {|P i}av

Diffusion Monte Carlo

In DMC the Metropolis method is used to directly sample the exact wave function where the MC walkers follow the dynamics of the imaginary time Schrödinger equation:

Consider the time dependent Schrödinger equation!!The formal time-dependent solutions are!!!!An arbitrary trial wave function is a linear combination of the time-independent eigenstates , which rotate in the complex plane with a frequency proportional to

i@

@t(x, t) = H (x, t)

(x, t) =1X

n=0

Cn n(x) exp(�iEnt)

H n(x) = En n(x)

n(x)En

Diffusion Monte Carlo

If we transform to imaginary time then!!!!As propagates in imaginary time the states all decay exponentially. Since excited states decay faster than the ground state, collapses to the ground state!

Diffusion Monte Carlo is a stochastic realisation of the imaginary time-evolution. is represented as a population of walkers that evolves with time according to the Hamiltonian

• Position representation : density of walkers maps to

• SD representation : density of walkers maps to in

@

@⌧(x, ⌧) = �H (x, ⌧)

⌧ = it

⌧

(x, ⌧)

(x, ⌧) =1X

n=0

Cn n(x) exp(�En⌧)

(x, ⌧)

(x, ⌧)

CP (⌧)

(⌧) =X

P

CP (⌧) P

Diffusion Monte Carlo

Compare!!!

The SE (I) is similar to the diffusion equation of a species A undergoing a first order reaction (II).

• is equivalent to the concentration of the species

• acts as a source or sink of species concentration

In FCI QMC, the equations become

!

Diagonal Hamiltonian elements act as the source or sink and the off diagonal elements diffuse the population of walkers

@

@⌧(x, ⌧) =

1

2r2 (x, ⌧)� V (x) (x, ⌧)

@[A]

@t(x, t) = Dr2[A](x, t)� k[A](x, t)

(I)

(II)

(x, ⌧)

V (x)

�⇥Ci

⇥�= (Hii � E0 � S)Ci +

X

j 6=i

HijCj

Summary

98+% of electronic structure calculations use DFT because it is cheap, often qualitatively accurate and implemented in many easy-to-use programs together with nuclear gradients and many other molecular properties.!When you want the right answer for the right reason, wave function methods can be used:

CCSD(T)-F12 for energies, gradients, molecular properties of molecules up to 30 atoms (linear scaling versions of this are under development for 200+ atoms)

CASSCF-based treatments for strongly correlated systems