Introduction to ab initio
methods I
Kirill Gokhberg
„We‘ve got fascinating results!“
„ ... this is ICD in the Ar dimer ...“
Anatomy of an ICD process
Resonant Auger
partial rates
Nuclear dynamics
in ICD states
ICD rates (R)
Core-Excited state PEC
Satellite Ar+*Ar PEC
Final Ar+Ar+ PEC
Initial vibrational WP
Optical excitation.
Dipole TM needed!
ICD electron
(and KER) spectra
Ar2 GS PEC
H  Te  T N  U ( r , R ) + relativistic terms if needed.
H describes the motion of N electrons and M nuclei.
2
2

pi
P
, TN  
 Te  
2
i
 2M 


Z
U ( r , R )  



i , | ri  R  |

 |r
i j
i
1
Electron interaction term
 rj |
H  H e  TN
He describes only electronic motion (nuclei fixed).
1. Bound electronic states  H e  n ( r , R )  V n ( R )  n ( r , R )
Kirill&Andreas
PEC or PES
Used to obtain the
„properties“ :TM, etc
2. Resonance (bound-in-continuum) states  V n ( R ),  n ( R )
Premysl
Electronic widths
3. Nuclear dynamics (in local approximation)
 (r , R ) 

n
( R ) n (r , R )
n
i


T

V
(
R
)


(
R
)

E
n (R) 
n
n
 N

2


Nicolas

nm
m (R)
m
time dependent formulation + coupling to final states
when computing ICD spectra
N-electron SE cannot be solved exactly!
Approximate solutions are found numerically and two
questions should be answered before starting the work.
1. What electronic structure method should we use?
2. How do we represent the respective Hilbert space
accurately enough?
or
What basis set should we select?
Choosing a method
• Independent particle (mean-field) methods – an
electron moves in an average field of (N-1) other
electrons.
Typical example - Hartree-Fock (HF) approximation.
• Correlated methods – a motion of an electron is
influenced by (correlated with) the motion of (N-1)
other electrons at each instant.
Examples – configuration interaction (CI), propagator (ADC),
many-body perturbation theory (MBPT), coupled cluster (CC), ...
methods.
• Hartree-Fock solution is used as an input for the correlated
methods.
Independent-electron wave function
spin-orbital
 ( x )   (r )   (  )
spin function
spatial orbital
Slater determinant
or
molecular orbital (MO) (ground state)
 1 
0  

 N !
 1/ 2
N!
 (  1)
pn
P n  1 (1 ) 2 ( 2 )
 N ( N )
n 1
•Electrons with anti-parallel spins are uncorrelated.
•Electrons with parallel spins are exchange
correlated.
Hartree-Fock approximation
E  0  
 0 H el  0
0 0
,
f (1 ) r (1 )   r r (1 )
f (1 )  h (1 )  J (1 )  K (1 )
E 0

J (1 ) r (1 )   
 i 1

*
1
d
x

(
2
)
r

(
2
)
 2 i 12 i   r (1 )

K (1 ) r (1 )   
 i 1

*
1
d
x

(
2
)
r

(
2
)
 2 i 12 r   i (1 )
N
N
HF equations
Coulomb and exchange operators
N
Total electronic energy
E HF 

ih i 
i 1
1
Coulomb and Exchange integrals
Orbital energy
Fock operator
ij r12 ij 
1
ij r12
ji 
1
N
N

2
i 1
1
1
ij r12 ij  ij r12
j 1
1
 d x 1 d x 2 i (1 ) i (1 ) r12  j ( 2 ) j ( 2 )
*
1
*
 d x 1 d x 2 i (1 ) j (1 ) r12  j ( 2 ) i ( 2 )
*
*
ji

Restricted vs. Unrestricted HF
• HF recovers up to 99% of the total electronic energy in the
ground state
 corr  E exact  E HF
• However, the energy differences of interest in chemical and
spectroscopic processes are a fraction of one percent of the
total energy.
• For example HF cannot describe binding between two rare-gas
atoms
Ar dimer electronic ground state
EHF=28670 eV, Eint [CCSD(T)]=11.5 meV
Koopmans‘ theorem
Negative of the HF orbital energies of the
occupied MOs are the electron binding energies
(ionization potentials).
IPi    i
Photoelectron spectrum of H2O
„Breakdown of the MO picture“
due to the intra-molecular correlation
Double ionisation threshold
Breakdown of monomer lines due to ICD
driven by inter-molecular correlation
1. MOs and orbital energies serve as input for the
correlated methods.
2. HF approximation furnishes us with a vivid picture
of many electron system with electrons stacked on
shelves called molecular orbitals.
3. HF solutions are of some, albeit limited, use for
computing electronic decay rates.
4. The HF approximation usually does not deliver
ground state PES in acceptable quality. It generally
fails to produce excited state PES at all.
5. It fails to reproduce correlation driven phenomena.
Choosing the basis set
M
i 
c
 ,A
i
  (r  R A )
 ,A
 STO  N r
n 1
 G TO  N x y z exp(   r )
exp(   r )Ylm ( ,  )
•Proper behaviour at r→0 and r→∞.
•Small number of STO basis functions
are sufficient to represent a MO.
l
m
n
2
•Allow for very efficient computation
of four-centre two-electron integals:
1
 A B r12  C  D 
 dx
1
d x 2  A (1 )  C (1 ) r12  B ( 2 ) D ( 2 )
*
1
*
cc-pVDZ basis set for Ne
Contracted Gaussian function
Primitive Gaussian function
 (r ) 
 c
i
i
( i , r )
i
Contraction coefficient
Contraction exponent
Core basis function
Valence basis function
Polarisation basis function
Adding diffuse functions
n
( diffuse )
  m ax 
(l )
n
RHF calculation of Ne
Augmenting a basis set
Introducing electron correlation
Configuration Interaction scheme

cˆ a − creation operator, cˆi − annihilation operator



 
Cˆ I  1ˆ ; cˆa cˆi ,  a , i ; cˆa cˆb cˆi cˆ j ,  a , b , i , j ; ...
Excitation (N electrons)

Cˆ I   cˆi ,  i ; cˆ a cˆi cˆ j ,  a , i , j ; ...
Ionisation (N-1 electrons)

Cˆ I   cˆi cˆ j ,  i , j ; cˆ a cˆi cˆ j cˆ k ,  a , i , j , k ; ...
Double ionisation (N-2 electrons)
 I  Cˆ I  0
H IJ   I H el  J
Diagonalisation

More on correlation!

Andreas‘s lecture tomorrow.
E0, E1, ... and corresponding wavefunctions