1 Computer   Experiment   12:   Electron   Correlation   in   the   Helium  

Atom  

1.1  Background  Electrons  move  in  a  molecule  in  a  very  complicated  way  because  they  experience  

attraction  from  the  nuclei  but  repulsion  from  the  other  electrons.  This  repulsion  

exists  due  to  the  classical  Coulomb  interaction  of  two  equally  charged  particles  plus  

a  pure  quantum  effect  –  an  exchange  ‘interaction’.  The  exchange  contribution  arises  

due  to  the  fact  that  two  electrons  can  not  exist  in  one  point  of  space  with  equally  

oriented  spin  (Pauli  exclusion  principle).  Therefore,  the  movement  of  electrons  is  

not  independent  and  the  movement  of  one  electron  influence  the  movements  of  all  

others.    

1.1.1 Hartree-­‐Fock  and  Electron  Correlation  The  Hartree-­‐Fock  method  describes  the  movement  of  an  electron  in  the  field  of  the  

nuclei  and  the  average  field  of  all  other  electrons.  This  is  a  surprisingly  good  

approximation  and  typically  yields  more  than  99.5%  of  the  correct  total  energy  of  a  

molecule.  However,  in  this  method  the  correlation  of  the  electron  (their  

“instantaneous”  interaction)  is  not  properly  accounted  for  and  consequently,  the  

remaining  errors  of  the  HF  approximation  are  still  very  large  on  a  chemical  energy  

scale  (several  hundred  kcal/mol).  Thus,  the  HF  method  can  not  provide  the  exact  

total  energy  and  is  usually  referred  to  an  “uncorrelated  approach”.1  As  a  result,  the  

correlation  energy  is  defined  as  the  difference  between  the  exact  total  energy  and  

the  energy  of  the  Hartree-­‐Fock  calculation:  

Ecorr= E

exact!E

HF                

 (1)  A  big  challenge  for  theoretical  chemistry  is  to  calculate  this  correlation  energy  to  

high  precision.  Unfortunately,  there  is  always  a  second  source  of  error  in  the  

calculations  which  stems  from  the  need  to  introduce  a  finite  basis  set  in  order  to  

                                                                                                               1  This   is   not   entirely   fair   since   the   HF   method   properly   includes   the   effects   of   the   “Fermi   correlation”   (thus,   the  exchange  correction  arising  from  the  Pauli  principle).  

represent  the  one-­‐electron  MOs  on  which  the  calculations  are  based.  It  is  obvious  

that  the  more  basis  functions  we  bring  in,  the  better  the  variational  principle  will  be  

able  to  model  the  shape  of  the  “real”  HF  MOs.  At  the  same  time  as  the  orbitals  

approach  the  true  HF  orbitals  in  the  limit  of  an  infinite  basis  set,  the  HF  energy  

approaches  the  basis  set  HF  limit.2  However,  in  practice  the  basis  set  limit  can  not  be  

reached  since  the  number  of  basis  functions  that  can  be  included  in  the  calculations  

is  limited  by  the  computational  time.3  At  the  same  time  the  error  of  introducing  a  

non-­‐complete  basis  set  –  the  basis  set  truncation  error  –  should  be  made  as  small  as  

possible.  In  the  limit  of  vanishing  error  one  obtains  results  in  the  basis  set  limit.  

Consider  for  example  a  series  of  HF  calculations  on  the  Ne  atom  with  basis  sets  of  

increasing  size  (energies  are  given  in  Eh):4  

 One  can  see  that  the  HF  limit  can  be  approached  closely  with  the  sufficiently  large  

basis  sets  and  for  the  chemical  accuracy  of  ~1  mEh    (with  respect  to  the  HF-­‐limit!)  

can  be  reached  for  the  TZV  and  QZV  basis  sets  which  are  already  small  enough  such  

that  they  can  be  used  in  molecular  calculations.  Note  however,  that  even  in  the  HF-­‐

limit  the  total  energy  is  still  far  from  the  exact  energy  due  to  the  missing  correlation.    

                                                                                                               2  Mathematically  speaking,  the  basis  set  should  approach  “completeness”.  3  In   fact,   the  computing   time   increases   (formally)  with   the   fourth  power  of   the  basis  set  size   (for  HF  and  worse   for  more  accurate  methods).  However,  there  are  also  issues  concerning  the  finite  accuracy  with  which  numbers  can  be  represented  by  a  digital  computer  which  also  puts  constraints  concerning  numerical  stability  on  the  size  of  the  basis  set  that  can  be  used.  4  The  details  of  how   these  basis   sets   come   to   their  names   is  not   important   in   the   context  of   this   course.   They  are  simply  basis  sets  of  increasing  size  and  accuracy.  

STO-3G : -126.60453 6s 3p (33/3) 3-21G : -127.80382 6s 3p (321/21) SV : -128.37641 7s 4p (511/31) TZV : -128.54149 11s 6p (62111/411) QZV : -128.54685 15s 9p (8211111/6111) Partridge-1 : -128.54695 14s 9p (uncontracted) Partridge-2 : -128.54708 16s11p (uncontracted) Partridge-3 : -128.54709 18s13p (uncontracted) HF-limit : -128.54710 Exact Energy (exp) : -129.056

1.1.2 Calculation  of  the  Correlation  Energy  In  order  to  recover  a  significant  fraction  of  the  missing  correlation  calculation,  we  

can  employ  a  more  complicated  expansion  of  the  wavefunction  by  adding  terms  

where  1,2,…,  N  electrons  in  the  HF  Slater  determinant  (i,j,k)  are  replaced  by  virtual  

orbitals  (a,b,c)  (excited  configurations).  Thus,  the  expansion  of  the  exact  N-­‐electron  

wavefunction  can  be  written  in  the  form:  

 

! =C0"

HF+ C

i

a"i

a

a

#i

#Singles

! "#### $####+

12!

$

%&&&&

'

())))

2

Cij

ab"ij

ab

a,b#

i,j#

Doubles

! "####### $#######+

13!

$

%&&&&

'

())))

2

Cijk

abc"ijk

abc

a,b,c#

i,j ,k#

Triples

! "######## $########+%   (2)  

This  way  of  writing  the  exact  N-­‐electron  wavefunction  is  known  as  the  “full-­‐CI”  

method.  The  number  of  terms  in  this  configuration  interaction  method  expansion  is  

extremely  large  and  –  in  fact  –  grows  in  factorial  way  with  the  number  of  electrons  

and  basis  functions.  Hence,  the  method  is  only  suitable  for  benchmark  calculations  

on  very  small  systems.  Using  todays  best  programs  and  computers,  CI  calculations  

can  be  (and  have  been)  done  with  up  to  ~1010  terms  in  the  determinantal  expansion  

within  the  restrictions  of  the  employed  one-­‐particle  basis  set.  The  N-­‐particle  

wavefunction  is  an  object  of  extreme  complexity  and  contains  all  information  that  

can  be  determined  on  the  quantum  system.    

1.1.3 Reduced  Density  Matrices  In  practice,  we  need  to  know  much  less  about  the  N-­‐particle  wavefunction  as  long  as  

we  are  only  interested  in  the  calculation  of  energies  and  observables.  In  fact,  the  

Born-­‐Oppenheimer  Hamilton  operator  only  contains  one  and  two-­‐body  terms.  Thus,  

the  exact  energy  can  already  be  calculated  from  the  so-­‐called  reduced  one-­‐particle  

and  two-­‐particle  density  matrices.  They  are  defined  as:  

! 1( ) x

1, !x

1( ) = N "* !x1,...,x

N( )" x1,...,x

N( )dx2...dx

N#      

  (3)  

! 2( ) x

1,x

2; !x

1, !x

2( ) = N N "1( ) #* !x1, !x

2,...,x

N( )# x1,x

2...,x

N( )dx3...dx

N$   (4)  

These  quantities  are  very  important  in  quantum  chemistry  since  they  contain  all  the  

necessary  information  for  calculating  the  total  energy  and  all  kinds  of  properties  of  

the  system.  They  called  here  “matrices”  assuming  that  two  arguments   x1  and   !x1

 

serve  as  continuous  „indices“  and  take  the  place  of  the  discrete  integer  indices  i  and  

j.  These  quantities  represent  generalizations  of  the  electron  density  function    

! 1( ) x

1( ) = N !" x1,...,x

N( )! x1,...,x

N( )dx2...dx

N#      

  (5)  

If  we  integrate  over  the  spin  of  the  first  electron,  we  obtain  the  spin-­‐free  first-­‐order  

density  function:  

P 1( ) r

1( ) = ! 1( ) r1s

1( )! ds1              

  (6)  

which  represent  the  probability  to  find  an  electron  at  the  space  part   r1  in  the  

infinitely  small  volume   dr1.  The  electron  density  is  observable  and  can  be  measured  

by  the  X-­‐ray  crystallography.  

Similar,  the  two-­‐electron  density  matrix  is  a  generalization  of  the  of  two-­‐body  

density  function:  

! 2( ) x

1,x

2( ) = N N !1( ) "* x1,x

2,...,x

N( )" x1,x

2...,x

N( )dx3...dx

N#  

  (7)  

Integrating  over  spins  of  the  first  and  the  second  electrons,  we  obtain:  

P 2( ) r

1,r

2( ) = ! 2( ) r1s

1,r

2s

2( )!! ds1ds

2            

  (8)  

The  last  expression  defines  the  probability  to  find  one  electron  of  an  arbitrary  spin  

at  the  point   r1  and  volume   dr1  with  any  spin  and  the  second  at  the  point   r2

 and  in  

the  volume   dr2  while  the  remaining  N-­‐2  electrons  can  have  arbitrary  positions  and  

spins.  Density  matrices  allow  a  very  useful  statistical  approach  to  the  analysis  of  the  

electron  correlation  which  is  not  related  to  the  total  energy.  

In  statistics  two  variables  x  and  y  with  probability  distributions  f(x)  and  g(y)  are  

uncorrelated  if  the  joint  probability  h(x,y)  is  given  by:  

h x,y( ) = f x( )g y( )                

  (9)  

For  our  case  this  implies  that  the  pair  density  should  be  a  simple  product  of  the  one  

particle  densities.    

P 2( ) r

1,r

2( ) = N N !1( ) 1N

P 1( ) r1( )

"#$$

%$$

&'$$

($$

1N

P 1( ) r2( )

"#$$

%$$

&'$$

($$=

N !1N

P 1( ) r1( )P 1( ) r

2( )   (10)  

1.1.4 Hole  Functions  In  order  to  obtain  a  parameter  of  the  correlation  “strength”  we  can  introduce  a  

quantity  which  defines  the  deviation  of  our  density  from  the  uncorrelated  case:  

P 2( )!! r

1,r

2( ) = P 1( )! r1( )P 1( )! r

2( ) 1+ f !! r1,r

2( ){ }        

 (11)  

P 2( )!" r

1,r

2( ) = P 1( )! r1( )P 1( )" r

2( ) 1+ f !" r1,r

2( ){ }        

  (12)  

The   f -­‐functions  called  the  parallel  and  antiparallel  spin  correlation  functions.  

The  probability  of  finding  the  first  electron  at   r1  with  the  condition  that  the  second  

electron  is  located  at   r2  with  the  same  spin  is  defined  by  

P 2( )!! r1,r

2( )P 1( )! r

2( ).  The  difference  

between  the  last  quantity  and  the  uncorrelated  probability  of  finding  the  first  

electron  at   r1  

hX

r1,r

2( ) =P 2( )!! r

1,r

2( )P 1( )! r

2( )!P 1( )! r

1( ) = P 1( )! r1( ) f !! r

1,r

2( )      

  (13)  

is  called  the  Fermi  hole.  In  the  same  way  one  can  define  similar  quantity  for  

antiparallel  spins:  

hC

r1,r

2( ) =P 2( )!" r

1,r

2( )P 1( )! r

2( )!P 1( )! r

1( ) = P 1( )! r1( ) f !" r

1,r

2( )      

  (14)  

which  is  called  the  Coloumb  hole.    The  idea  of  the  Fermi  and  Coulomb  holes  allow  a  

very  pictorial  and  intuitive  approach  for  understanding  of  how  exchange  and  

Coulomb  interaction  affect  the  electron  distribution  in  an  atom  or  molecule.  One  can  

image  an  electron  digging  a  hole  around  so  that  the  probability  to  find  another  

electron  nearby  is  diminished.  The  form  of  the  Fermi  hole  is  generally  known  only  

for  a  few  cases  such  as  a  uniform  electronic  gas,  but  at  the  short  electronic  distances  

the  hole  behaves  as   r122 .  The  shape  of  the  Coulomb  hole  function  is  complicated  and  

not  well  known.  However,  it  is  known  that  at  the  small  electronic  distances  the  hole  

function  has  a  cusp.  

1.1.5 Natural  Orbitals  The  one-­‐electron  density  matrix  can  be  written  for  a  complete  set  of  spin-­‐orbitals  

as:  

! 1( ) x

1, !x

1( ) = !pq

1( )"p

x1( )"q

* !x1( )

p,q"            

  (15)  

Since  a  unitary  transformation  applied  for  a  set  of  spin-­‐orbitals  leaves  the  wave  

function  unchanged,  we  can  transform   !

pq

1( )  to  the  diagonal  form:  

U+! 1( )U = n =

n1

0 ! 0

0 n2! 0

" " # "0 0 ! n

!" 0

#

$

%%%%%%%%%%%%%

&

'

(((((((((((((((

np

p) = N    

  (16)  

The  numbers  ni  are  the  occupation  numbers  of  the  natural  spin  orbitals.  They  

can  be  shown  to  be  0  ≤  np  ≤  1.  They  are  usually  arranged  in  order  of  decreasing  

occupation  number.  The  natural  spin  orbitals  are:  

!1

NSO( )

!2

NSO( )

!

!!

NSO( )

"

#

$$$$$$$$$$$$$$

%

&

'''''''''''''''''

= U

!1

!2

!!!

"

#

$$$$$$$$$$$$$

%

&

'''''''''''''''

             

  (17)  

The  NOs  usually  come  as  N  orbitals  with  an  occupation  number  close  to  1  (strongly  

occupied  NSOs)  and  the  remaining  NSOs  with  small  occupation  numbers  (weakly  

occupied  NSOs).  The  first  order  density  becomes:  

! 1( ) x

1, !x

1( ) = np"

p

NSO( ) x1( )"

p

NSO( )* !x1( )

p

"          

  (18)  

Similarly  one  can  obtain  a  set  of  orbitals  which  diagonalize  the  spin-­‐free  first-­‐order  

reduced  density  matrix.  They  are  called  Natural  Orbitals  (NOs):  

P

1( ) r1, !r

1( ) = np!

p

NO( ) r1( )!

p

NO( )* !r1( )

p

"             (19)  

The  occupation  numbers  here  are  confined  to  0  ≤  np  ≤  2.  An  important  theorem  

states  that  CI  expansions  built  on  natural  orbitals  show  the  fastest  possible  

convergence.  That  is,  the  natural  orbitals  with  negligible  occupation  numbers  can  be  

omitted  from  the  one-­‐electron  space  with  negligible  consequence  on  the  accuracy  of  

the  CI  expansion.    

For  example,  the  full-­‐CI  calculations  on  the  He-­‐atom  with  the  large  aug-­‐cc-­‐pV6Z(-­‐h)  

basis  set  (Figure  1)  show  that  the  convergence  initially  is  very  rapid.  According  to  

this  picture,  only  a  few  orbitals  (up  to  4p  which  is  about  18  correlating  orbitals  for  

this  electron  pair)  need  to  be  included  in  the  correlation  treatment  in  order  to  reach  

chemical  accuracy  (~1  mEh)  in  the  correlation  energy.    

HF 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5d 5f 5p 5g 5s all

-0,040

-0,030

-0,020

-0,010

0,000

Chemical AccuracyReached (1 kcal/mol)

Tota

l Ene

rgy

(Eh)

-E(H

F=--

2.86

1673

13)

NOs included in the CI

Exact Correlation Energy

 Figure  1:  Full-­‐CI  calculations  on  the  He  atom  in  a  basis  of  natural  orbitals  derived  from  the  aug-­‐cc-­‐pV6Z(-­‐h)  basis  set.   Plotted   is   the   error   in   the   calculation   made   by   only   including   the   indicated   NOs   on   the   x-­‐axis   in   the   CI  expansion.    

 

1.2  Description  of  the  Experiment  In  this  experiment  you  will  carry  out  a  few  full-­‐CI  calculations  on  the  He-­‐atom  –  the  

simplest  two-­‐electron  system.  In  this  case  a  CI  calculation  with  single-­‐  and  double  

excitations  already  represents  the  full-­‐CI  problem  and  we  concentrate  on  studying  

the  basis  set  convergence  of  the  HF  energy  and  the  correlation  energy.  

JOB:  • Study   the   He-­‐atom   with   the   cc-­‐pVXZ   (X=2,3,4,5,6)   basis   sets,   the   ORCA  

program  and  the  MDCI  module.  The  input  for  the  ORCA  program  can  be  used  

as  follows:  

 Run  calculations  using  the  basis  sets  cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z, cc-pV6Z.

# Full CI He calculation ! RHF cc-pVDZ TightSCF %mdci CIType CISD Ewin -4,1e10 End * xyz 0 1 He 0.0 0.0 0.0 *

• Plot   the   correlation   energy   as   a   function  of  X   and  observe   its   convergence.  

Estimate  the  basis  set  limit.  In  order  to  accomplish  this  proceed  as  follows:  

Make  an  input  file  energy.dat  in  the  form  of  2 Correlation energy cc-pVDZ 3 … 4 … 5 … 6 Correlation energy cc-pV6Z

For  visualisation  with  xmgrace  type:   xmgrace –legend load energy.dat

The  relation  between  the  correlation  consistent  energies  and  the  energy  of  

the  basis  set  limit  may  be  written  as    

Eexact! E

X+ AX"3              

  (20)  

where  X  denotes  the  correlation  consistent  basis  set  cc-­‐pVXZ  and   EXis  the  

correlation  energy  calculated  with  the  basis  set  cc-­‐pVXZ.  Now  calculating  two  

energies   EX  and   EY

with  the  basis  sets  X  and  Y  we  have  

Eexact

= EX

+ AX!3

Eexact

= EY

+ AY !3              

  (21)  

Solving  this  for   Eexactwe  obtain  the  following  expression  for  the  extrapolated  

correlation  energy  and  the  parameter  A:  

Eexact

=X 3E

X!Y 3E

Y

X 3!Y 3

A =!E

X!E

Y

X!3!Y !3

           

  (22)  

Calculate  the  basis  set  limit  for  the  correlation  energy  for  X=2,  Y=3.  

• Plot  the  SCF  energy  as  a  function  of  X  and  observe  its  convergence.  Estimate  

the  basis  set  limit.  Using  the  same  procedure,  it  is  possible  to  extrapolate  the  

total   energy   to   the   basis   set   limit.   Having   done   two   calculations   with   the  

basis  sets  cc-­‐pVXZ  and  cc-­‐pVYZ  with  X<Y  we  obtain:  

E

exact

tot = EX

HF +X

3E

X

corr !Y3E

Y

corr

X3!Y

3          

  (23)  

Calculate  the  exact  total  energy  for  X=2  and  Y=3.  Compare  your  result  to  the  

accurate  CCSD(T)-­‐R12  calculation  of  Klopper  of   E

r12!CCSD(T )tot =!2.9037061 E

h

and   E

r12!CCSD(T )corr =!42.037 mE

h.    

• Determine   an   experimental   total   energy   of   the   He   atom.   This   can   be  

calculated   as   a   sum   of   ionization   potentials  

E

totHe( ) = I He! He

+( )+ I He+ ! He

2+( ) .   From   the   NIST   tables   the   first  

ionization   potential   is   I He! He

+( ) = 24.5874 eV .   The   second   ionization  

potential  can  be  obtained  as   the  energy  of   the  hydrogen-­‐like  atom  with   the  

charge   2   using   the   fact   that   the   energy   of   the   hydrogen-­‐like   atom   scales  

quadratically  with  respect  to  the  nuclear  charge.  

• Determine   the  natural  orbitals  of   the   system   for   the   largest  basis   set.  Redo  

the  calculation  with  limited  set  of  natural  orbitals  and  observe  which  fraction  

of   the  correlation  energy   is  recovered  after   terminating  the  CI  at   the  2s,  3s,  

3p,  3d,  4s,  4d,  4f,  5s,  5p,  5d,  5f,  5g  …  natural  shell.    

Natural  orbitals  are  stored  in  the  file  *.mdci.nat.  They  can  be  loaded  from  

file  with  the  options  moread and moinp  

 

# load natural orbitals ! RHF cc-pVDZ TightSCF ! moread noiter %moinp “orbitals.mdci.nat” * xyz 0 1 He 0.0 0.0 0.0 *

Look  into  the  orbital  energies  and  determine  1s,  2s,  2p  etc.  natural  shells.  The  

orbitals  included  into  CI  calculation  can  be  selected  by  adjusting  the  

parameter  ewin (=orbital  window)  in  the  mdci  block.  

• Do  radial  plots  of  the  1s,  2s,  3s,  4s,  5s  natural  orbitals.  Is  there  are  regularity?  What   do   you   think   you   describe  with   these   natural   orbitals?  What   are   the  

higher  angular  momentum  NOs  good  for?  Can  you  distinguish  several  types  

of  correlation  that  are  described  with  different  NOs?  Compare  the  1s  and  2s  

natural  orbitals  to  the  1s  (occupied)  and  2s  (unoccupied)  HF  orbitals.  What  

do  you  observe?  Are   the  virtual  HF  orbitals  a  suitable  basis   for   introducing  

correlation?    

Produce  the  data  file  containing  the  1s,  2s,  3s,  4s  and  5s  natural  orbitals  of  He  

using  the  following  input  file  for  ORCA:  

 The  resulting  data  file  “He-­‐nat1s.dat”  is  a  standard  two-­‐column  ASCII  file  and  

can  be  plotted  with  any  plotting  program  like  xmgrace  or  gnuplot.  Plot  the  

radial  distribution  functions  for  1s,  2s,  3s,  4s  and  5s  orbitals  in  the  form  of  

! r( )

2r 2 .  

# Plot 1s natural orbital of He ! RHF cc-pV6Z TightSCF ! moread noiter %moinp "orbitals.mdci.nat" %plots dim1 512 dim2 1 min1 0 max1 10 min2 0 max2 1 Format origin; MO("He-nat1s.dat",0,0); v1 0,0,0; v2 0,0,1; v3 0,1,0; end * xyz 0 1 He 0.0 0.0 0.0 *

2 Computer   Experiment   13:   Potential   Energy   Surfaces   Using  Correlated  ab  initio  Methods  

2.1 Background  In  this  computer  experiment  we  are  going  to  calculate  the  potential  energy  surfaces  

of  some  very  small  diatomic  molecules  using  multireference  ab  initio  methods.  It  is  

seen  how  the  geometric  and  electronic  structure  of  the  molecules  changes  from  one  

state  to  the  other  which  will  be  related  to  the  bonding  in  the  molecule.  Secondly,  it  

will  be  studied  how  the  electronic  structure  in  the  equilibrium  region  correlates  

with  the  free-­‐atom  dissociation  products.  Thirdly,  it  will  be  studied  how  rigorous  

multireference  ab  initio  methods  achieve  comparatively  high  accuracy  in  the  

prediction  of  the  electronic  structure  of  small  molecules.5  

2.1.1 Potential  Energy  Curves  of  Diatomic  Molecules  In  diatomic  molecules  there  is  only  a  single  degree  of  freedom  –  the  internuclear  

separation.  Therefore,  the  potential  energy  curves  can  be  readily  vizualized  and  a  lot  

of  insight  into  the  electronic  structure  of  the  electronic  states  being  studied  can  be  

gained.  We  will  not  enter  a  detailed  discussion  of  the  theoretical  background  here  

and  refer  to  the  classical  text  of  Herzberg  for  this  purpose.6  In  these  books  the  

theory  is  discussed  in  detail  on  an  accessible  level.  In  addition,  Herzerg’s  books  

provide  a  comprehensive  overview  of  the  experimental  data  available  at  the  time  and  

provides  an  invaluable  resource  for  anybody  who  performs  calculations  on  diatomic  

molecules.  

In  the  Born-­‐Oppenheimer  approximation,  the  energy  of  a  given  rovibrational  state  

of  a  diatomic  molecule  nicely  separates  into  contributions  from  the  electronic  

energy,  the  vibrational  energy  and  the  rotational  energy.7  For  a  given  electronic  

state  I  it  reads:  

                                                                                                               5  Again,  we  will  use  basis  sets  here  that  are  smaller  than  would  be  „state-­‐of-­‐the-­‐art“.  Consequently,  the  full  potential  of  the  ab  initio  methods  will  not  be  realized  here.    6  Herzberg,  G.  Molecular  Spectra  and  Molecular  Structure.  I.  Spectra  of  Diatomic  Molecules.  Van  Nostrand,  New  York,  1950;  Huber,  K.P.;  Herzberg,  G.;  Molecular  Spectra  and  Molecular  Structure.  IV  Constants  of  Diatomic  Molecules.  Van  Nostrand,  New  York,  1979.  7  There  are  even   finer   contributions  coming   from  relativistic  effects,   the  electron   spin  as  well   as   the  hyperfine  and  quadrupole   interactions   that   were   partly   studied   in   chapter   Error!   Reference   source   not   found.   and   will   be  

 

EI

R,!,J( ) = Eel

I( ) + Evib

I( ) + Erot

I( )

!Te

I( ) R( )+G I( ) !( )+ F I( ) !,J( )           (24)  

The  electronic  contribution  Te  is  the  “raw”  electronic  energy  that  we  are  trying  so  

desperately  to  calculate  to  good  accuracy  with  our  electronic  structure  methods.  

The  vibrational  energy  function  G  is  approximately  represented  by:  

  G I( ) !( ) = "

e

I( ) ! + 12( )!"e

I( )xe

I( ) ! + 12( )2

+ "e

I( )ye

I( ) ! + 12( )3

+ ...    

  (25)  

Where  ν=0,  1,  2,  …  is  the  vibrational  quantum  number.  Thus,  in  general  it  contains  

beyond  the  harmonic  vibrational  energy   !e

I( )  (which  changes  from  state  to  state),  

corrections  for  „anharmonicities“  that  are  described  by   !e

I( )xe

I( ),!e

I( )ye

I( ) .  The  

rotational  energy  is  given  by:  

  F I( ) !,J( ) = B

!

I( )J J +1( )!D!

I( )J 2 J +1( )2+ ...        

  (26)  

(J  =  0,1,2,…).  Here   B!I( ),D

!

I( )  are  the  rotational  constants  that  depend  –  in  general  –  

on  the  electronic  and  vibrational  states  of  the  system.  For  our  purposes,  the  

contributions  of  the  rotations  are  so  small  that  they  will  be  neglected.  In  high  

resolution  gas  phase  spectroscopy  transition  between  the  different  vibronic  states  

of  the  system  are  observed  in  either  absorption  or  fluorescence.  From  the  data  one  

can  derive  –  in  principle  –  a  number  of  spectroscopic  constants:  

• The  symmetry  of  a  given  state.  

• The  equilibrium  distance   RI( )  for  each  electronic  state  I.  

• The  vibrational  frequency   !e

I( )  for  each  electronic  state  I.  

• The  anharmonicities   !e

I( )xe

I( ),!e

I( )ye

I( )  for  each  electronic  state  I.  

                                                                                                                                                                                                                                                                                                                                         disregarded  here.  There  are  also  various  couplings  between  the  different  interactions  that  we  will  also  ignore  for  the  present  purposes.  

Thus,  relative  to  the  ground  state  one  can  determine:  

• The  adiabatic  excitation  energy   Te

I( )  for  each  excited  electronic  state.  This  is  

neither   the   energy   of   the   0-­‐0   transition8  nor   the   energy   of   the   vertical  

transition 9  but   simply   represent   the   difference   in   energy   between   the  

minima  of  the  two  curves.  It  is  given  by:   T

e

I( ) R I( )( )!Te

0( ) R 0( )( ) .  

• The   shift   of   the   equilibrium  distance   RI( )!R 0( ) .   This   is   the  most   important  

quantity  in  determining  the  intensity  distribution  in  the  vibronic  spectra.10  

A  typical  set  of  potential  energy  surfaces  (PESs)  are  shown  in  Figure  2  for  the  C-­‐H  

molecule.  While  the  calculations  leading  to  these  curves  have  been  slightly  

oversimplified  they  already  show  some  important  features:  

• The  shapes  of  the  potential  energy  surfaces  can  be  quite  complicated.  In  addition  

to   the   minima   in   the   “quadratic   region”   near   R=1.1   Angström,   there   are  

crossings  of  curves,  local  maxima  and  related  phenomena.  

• The  curves  “re-­‐unite”  at  “infinite”  distances  to  a  combination  of  the  states  of  the  

free  atoms.    

• The   symmetry   of   each   state   is   determined   by   inspection   of   the   orbital  

configurations   that   dominate   the   electronic   structure   of   the   state   (vide   infra)  

together  with  group  theoretical  tables.    

• The  equilibrium  distance  is  different  for  each  state  and  it  is  already  obvious  from  

inspection  that  the  curvature  around  the  minimum  changes  from  one  state  to  the  

other.  

                                                                                                               8  The  0-­‐0  transition  is  the  energy  for  a  transition  between  the  two  lowest  vibrational  levels  (ν=0)  of  the  two  electronic  states  involved  in  the  transition.  9  The   „vertical   transition“   is   the   electronic   energy   difference   between   the   two   vibronic   states   at   the   equilibrium  distance  of  the  electronic  ground  state  (or,  more  generally,  the  initial  state  involved  in  the  transition).  10  It   is   of   course   also   possible   to   study   the   electric,   magnetic   and   quadrupolar   transition   probabilities   between  vibronic  levels.  We  will  not  pursue  this  here.  

• The   potential   energy   surfaces   were   calculated   pointwise.   Thus,   the   electronic  

structure  problem  was  solved  on  successive  internuclear  distances  and  a  smooth  

curve   is   obtained   from   interpolation.   This   is   characteristic   of   the   Born-­‐

Oppenheimer   approach   –   we   approximately   solve   the   electronic   Schrödinger  

equation  for  each  fixed  nuclear  arrangement  separately.  

1.0 1.5 2.0 2.5 3.0 3.5

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-38.32

-38.30

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-38.26

-38.24

-38.22

-38.20

-38.18

-38.16

-38.14

0

5000

10000

15000

20000

25000

30000

35000

40000

Rel

ativ

e E

nerg

y (c

m-1)

C(1D)+H(2S)

C(3P)+H(2S)

C-2Σ+

B-2Σ−

A-2Δ

a-4Σ

Tota

l Ene

rgy

(Eh)

C-H Distance (Angström)

X-2Π

 Figure  2:  The  potential  energy  surfaces  of  the  low-­‐lying  excited  states  of  the  C-­‐H  molecule  as  calculated  with  the  MRCI+Q  method  and  the  SVP  basis  set.    

Suppose  that  we  have  calculated  a  number  of  such  potential  energy  surfaces   E

el

I( ) R( )  and  we  whish  to  determine  the  spectroscopic  constants.  How  can  you  do  that?  

Below  is  a  simple  recipe:  

• First,  a  smooth  curve  is  fitted  through  the  available  points.  In  practice  this  may  

be  a  cubic  spline  or  Chebyshev  polynomials  or  similar  techniques.  

• Second,  a  search  for  the  minima  on  each  potential  energy  surface  is  carried  out.  

These   are   found  by   examing   triples   of   points   on   the  PES   surface.   If   one  of   the  

three  points  is  lower  in  energy  then  the  two  neighbouring  points  to  the  left  and  

the  right,  there  must  be  a  minimum  close  by.  This  may  then  be  located  precisely  

using  the  interpolated  curves  and  standard  techniques  of  numerical  analysis.  

• Once  the  equilibrium  distance  is  found,  the  second  derivative  of  the  interpolated  

curve   is   calculated.   From   the   value   of   !2E

el

I( ) /!R2

R=RI( ),   (in   Eh/(au2))   and   the  

reduced  mass  of  the  molecule  ( µ = M

AM

B/ M

A+ M

B( ) ,  in  Dalton)  the  harmonic  vibrational  frequency  (in  wavenumbers)  is  calculated  by:11  

! I( ) =

12"c

!2E!R2

R=RI( )

1µ

= 5140.487!2E!R2

R=RI( )

1µ      

  (27)  

• From   the   value   of   the   electronic   energy   at   the   equilibrium  distance,   the   “term  

energy”  Te   is  determined.  Subtraction  of   the  ground  state  Te   and  conversion   to  

wavenumber  units12  results  in  a  number  that  can  be  compared  with  Herzberg’s  

tables.    

• From   the   difference   in   equilibrium   distance   relative   to   the   ground   state  

RI( )!R 0( )  (in   atomic   units),   the   dimensionless   displacement   parameter   !

I( )  is  

calculated:  

! I( ) =!R

2!cµ" I( )

h= 0.091139!R µ" I( )      

 (28)  

( !R  in  atomic  units,   µ  in  Dalton  and   !I( )  in  cm-­‐1)  These  dimensionless  

displacements  are  quite  useful.  First  of  all  they  define  the  so-­‐called  Huangh-­‐Rhys  

factor   S I( ) = 1

2! I( )( )

2

.  In  terms  of  these  parameters  the  Franck-­‐Condon  factors  for  

vibrational  progressions  can  be  given  in  closed  form  provided  that  both  PESs  are  

assumed  to  be  harmonic  –  this  is  a  rough  approximation  but  a  useful  first  

                                                                                                               11  The  conversion  is  a  little  painful.  You  want  to  give  the  energy  second  derivative  in  Eh/(au2)  and  the  reduced  mass  in  Dalton.   The   best   way   is   to   convert   everything   to   SI   units   and   finally   back   to   wavenumbers.   The   units   reduce   as  follows:  1  J  =  1  Nm  =  1  m2/s2  kg  so  J/m2*1/kg  =  m2/s2  kg/m2*1/kg=1/s2.  Thus,  the  prefactor  has  the  unit  s/m  and  the  integral  reduces  to  1/s  and  the  final  prefactor  is  1/m  =  1/100*cm-­‐1  12  Using  1  au  =  219474  cm-­‐1  

orientation.  For  transitions  out  of  the  lowest  vibrational  level  of  the  ground  state  

into  the  m’th  vibrational  level  of  the  excited  states  PES  the  FC  factors  are  given  by:  

    0 m m 0 =

S I( )( )m

m !e!S

I( )            

  (29)  

The  maximum  intensity  in  the  progression  will  occur  roughly  for  m  ≈  S.    

An  alternative  to  the  polynomial  fitting,  is  a  fit  to  a  Morse  potential  curve.  Such  a  

more  restricted,  parameterized  form  has  an  advantage  if  the  calculated  data  contain  

some  numerical  noise  (as  is  the  case  with  the  MRCI  variant  that  we  will  use  below).  

The  Morse  potential  is:13  

  E

MorseR( ) = E

0+ D

e1!e!! R!R( )( )

2

           

  (30)  

 Figure  3:  The  Morse  function  and  its  parameters.  

The  Morse  function  has  the  following  derivatives  at   R = R :  

 

!EMorse

!RR=R

= 0                

  (31)  

                                                                                                               13  An   excellent   short   discussion   is   given   by   A.C.   Hurley:   Introduction   to   the   Electron   Theory   of   Small   Molecules,  Academic  Press,  London,  New  York,  1976,  chapter  1.  

 

!2EMorse

!R2

R=R

= 2!2De              

  (32)  

 

!3EMorse

!R3

R=R

="6!3De              

  (33)  

 

!4EMorse

!R4

R=R

= 14!4De              

  (34)  

The  Morse  function  has  four  fit  parameters:   E0,D

e,!,R .  After  these  have  been  

determined  by  least-­‐square  fit  to  the  calculated  surface,  the  harmonic  frequency  is  

calculated  from  (27)  and  (32)  while  the  anharmonic  constant  is  given  by:  

  !

ex

e=

h"2

8#2µc= 60.16623

"2

µ[cm!1 ]          

  (35)  

Here,   De  is  Eh,   !  is  in  au

-­‐1  and   µ  is  in  Dalton.      

Finally,  the  most  accurate  method  is  to  directly  solve  for  the  vibrational  energy  

levels  of  a  given  PES.  In  this  case,  one  can  write  the  unknown  vibrational  

wavefunction  as  a  series  expansion  in  harmonic  oscillator  wavefunctions  and  apply  

the  variational  principle  in  order  to  find  the  vibrational  eigenstates  in  an  arbitrary  

potential.  The  required  integrals  can  either  be  calculated  by  numerical  integration  

techniques  or  may  also  be  obtained  in  closed  form  by  assuming  certain  shapes  for  

the  PES.  If  the  harmonic  oscillator  basis  is  large  enough,  one  obtains  accurate  results  

for  vibrational  energy  levels  as  well  as  vibrational  eigenfunctions  that  can  then  be  

used  to  calculate  accurate  Franck-­‐Condon  factors.  

2.1.2 Single  reference  ab  initio  Methods  As  explained  in  chapter  1  (page  1),  the  Hartree-­‐Fock  method  provides  in  many  cases  

a  reasonable  0th  order  description  of  molecules.  As  “the”  HF  method  we  interprete  

methods  in  which  a  single  determinant  is  taken  as  Ansatz  for  the  N-­‐electron  

wavefunction  and  the  energy  of  this  determinant  is  minimized  through  variation  of  

the  orbitals  in  order  to  achieve  the  lowest  possible  energy.  As  described  below,  

there  are  molecules  and  states  that  can  not  be  described  to  0th  order  by  a  single  

determinant.  For  these  molecules  and  states  one  must  use  the  “multireference  

methods”  described  below.  For  a  large  class  of  molecules,  in  particular  for  many  

closed-­‐shell  molecules  close  to  their  equilibrium  geometry,  the  single  determinantal  

HF  wavefunction  is  a  reasonable  approximation.  What  remains  to  be  done  in  order  

to  improve  on  the  HF  approximation  up  to  the  point  where  it  gives  quantitative  

agreement  with  experiment,  is  to  take  care  of  the  dynamic  correlation.14  One  way  

to  approach  this  problem  is  to  pass  from  a  single  determinant  Ansatz  to  a  many  

determinant  Ansatz  for  the  N-­‐particle  wavefunction.  However,  one  should  choose  

the  many  determinants  in  the  expansion  in  a  systematic  fashion.  The  most  popular  

of  the  systematic  approaches  is  to  arrange  the  possible  determinants15  by  excitation  

level.  As  an  “excitation”  one  does  not  understand  an  actual  excited  state  of  the  

system  but  simply  the  construction  of  determinants  in  which  occupied  HF  orbitals  

(labels  i,j,k)  are  replaced  with  virtual  orbitals  (a,b,c).  The  exact  N-­‐particle  

wavefunction  can  then  be  written  as:  

  ! = !

HF+ C

a

i!i

a

ia

" + 14

Cab

ij!ij

ab

ijab

" + 136

Cabc

ijk!ijk

abc

ijkabc

" + ...      

  (36)  

The  different  terms  represent  the  HF  “reference”  determinant,  single-­‐excitations,  

double  excitations,  triple-­‐excitations  and  “…”  indicates  that  this  scheme  is  to  be  

extended  up  to  N-­‐tuple  excitations.16  It  is,  unfortunately,  highly  impractical  to  base  

actual  calculations  on  such  a  “brute”  force  scheme  since  it  is  evident  that  the  

number  of  terms  in  the  expansion  grows  so  quickly  that  even  the  largest  computers                                                                                                                  14  Essentially  the  notion  that  the  movement  of  the  electrons  depend  in  a  very  complicated  way  on  the  movements  of  the  other  electrons.  In  the  Hartree-­‐Fock  methods,  each  electrons  just  “swims”  in  an  average  potential  “sea”  created  by  the  remaining  N-­‐1  electrons.  15  There   are   on   the   order   of   N!   possible   determinants   in   a   N-­‐electron   system   which   immediately   leads   to   the  conclusion  that   it   is   impossible   to   introduce  them  all   in  the  calculation  (except   for  the  smallest  benchmark  systems  where  this  has  been  extensively  pursued).  16  One   useful   way   to   think   about   the   configuration   expansion   coefficients   C   is   that   they   (actually   their   square)  describe  the  probability  of  electrons  to  temporarily  “jump”  out  of  their  HF  orbitals  and  partly  occupy  virtual  orbitals.  These  jumps  reduce  the  electron-­‐electron  repulsion  since  since  the  average  electron-­‐electron  distance  is  larger  if  the  electrons  occupy  different  orbitals.  

and  most  efficient  programs  can  not  handle  them.  Thus,  the  number  of  singles  is  

proportional  to  O(N2),  the  number  of  doubles  to  O(N4),  the  number  of  triples  to  

O(N6)  etc.  Fortunately,  the  importance  of  the  excitation  classes  diminish  with  

increasing  excitation  level.  The  by  far  most  important  effects  are  brought  in  by  the  

double  excitations,17  the  triple-­‐excitations  provide  important  refinements  and  the  

quadruple  excitations  are  only  important  in  special  contexts.18  Since  the  number  of  

doubles  is  “only”  proportional  to  O(N4)  chemically  significant  calculations  can  be  

done  for  up  to  double  excitations.19  The  treatment  of  triple-­‐excitations  is  restricted  

to  small  molecules  and  in  an  approximate  way  for  at  most  medium  sized  

molecules.20    

The  different  classes  of  single-­‐reference  correlation  methods  mainly  differ  in  the  

way  that  the  expansion  coefficients   Cai ,   Cab

ij ,   Cabcijk ,…  in  eq  (36)  are  determined.  One  

possibility  is  to  approach  the  problem  by  perturbation  theory.  In  this  case  the  

complete  Born-­‐Oppenheimer  Hamiltonian  is  divided  into  a  0th  order  part  which  has  

the  HF  solution  as  its  eigenfunction  and  the  remainder   V = H BO ! H 0( )  (sometimes  

called  the  “fluctuation  potential”).21  Such  methods  are  quite  successful  and  can  be  

implemented  and  applied  roughly  up  to  the  4th  order  in  perturbation  theory.  

However,  electron  correlation  is  not  a  small  perturbation  and  failures  of  these  

methods  due  to  a  lack  of  convergence  of  the  perturbation  expansion  are  not  

uncommon.  Very  popular  is  the  2nd  order  method  (MP2)  which  represents  a  major  

                                                                                                               17  The  singles  to  contribute  very  little  to  the  correlation  energy  due  to  “Brillouin’s  theorem”.  18  The   so-­‐called   “disconnected”   quadruples   restore   the   size-­‐consistency   of   the   CISD   expansion   which   is   of   major  importance.  These  excitations  may  be   thought  of  as   independent  pair  excitations.   “Genuine”  quadruple  excitations  contribute  very  little  to  the  correlation  energy.  19  However,   in  practice  we  need   to  compute   the   interactions  between  double  excitations.  While   there  are   formally  O(N8)  doubles/doubles  interactions,  “only”  O(N6)  of  them  are  nonzero.  Thus,  methods  that  require  the  calculation  of  doubles/doubles  interactions  scale  as  O(N6)  unless  further  approximations  are  introduced.  This  puts  severe  limits  on  the  size  of  the  molecules  that  can  be  treated  with  such  methods.    20  This   may   change   with   the   advent   of   the   now   emerging   “linear   scaling   local   correlation”   methods.   Here   the  calculations   are   arranged   such   that   terms   of   negligible   size   can   be   identified   and   neglected.   The   remaining  “significant”  contributions   to   the  correlation  energy  only   scale   linearly  with   the  size  of   the  molecule   since  electron  correlation  is  a  rather  short-­‐range  phenomenon.  21  There  are  many  ways  in  which  “reasonable”  definitions  of  H(0)  and  V  can  be  accomplished.  The  most  popular  choice  is  the  Möller-­‐Plesset  (MP)  choice  in  which  H(0)  is  the  sum  of  the  Fock  operators  for  the  N-­‐electrons.  

improvement  over  HF  theory  and  has  a  comparatively  low  cost  (proportional  to  

O(N5)).22    

Methods  that  are  of  infinite  order  in  the  fluctuation  potential  are  frequently  the  

methods  of  choice  since  they  do  not  assume  smallness  of  electron  correlation  

effects.  However,  straightforward  application  of  the  variational  principle  to  an  

expansion  that  is  truncated  at  the  level  of  single  and  double  excitations  (called  

CISD)  is  not  a  successful  method  because  it  has  an  important  defect:  it  lacks  size  

consistency.  That  is,  for  noninteracting  fragments  A  and  B  a  calculation  of  the  

supermolecule  AB  yields  a  different  result  than  the  sum  of  calculations  on  the  

isolated  fragments  A  and  B.  Thus,  the  method  behaves  inconsistently  as  the  size  of  

the  molecule  changes  and  this  is  deadly  for  its  application  to  chemistry.    

The  size  consistency  is  restored  by  including  higher  excitations  but  their  full  

inclusion  is  impractical  (as  pointed  out  above).  However,  the  situation  is  not  so  bad  

since  in  order  to  “repair”  the  size-­‐inconsistency  of  the  CISD  method  one  “only”  

needs  to  approximately  incorporate  the  doubles/quadruples  interaction.  Of  key  

importance  are  the  quadruple  excitations  that  can  be  described  as  products  of  

double  excitations  (“simultaneous  pair  excitations”).  Their  coefficients  are  

essentially  given  by  products  of  double  excitation  coefficients.  The  earliest  attempts  

to  include  this  important  excitation  class  are  known  as  “coupled  electron  pair”  

models  (CEPA).23  These  methods  are  size  consistent,  but  no  longer  variational  (as  

the  CISD  method).  The  size-­‐consistency  is  the  more  important  property!    

The  CEPA  methods  make  some  approximations  that  prevent  them  from  including  

the  disconnected  quadruples  rigorously.24  The  simplest  method  that  does  so  is  the  

                                                                                                               22  Very   efficient   and   accurate   approximations   to   the   exact  MP2  method   exist.   They   lead   to   calculations  where   for  most   molecules   of   practical   size   (~1000   basis   functions),   the   initial   HF   calculation   takes   more   time   than   the  computation  of  the  MP2  correlation  energy.  23  W.  Meyer  Int.  J.  Quant.  Chem.  (1971),  5,  341;  W.  Meyer  J.  Chem.  Phys.  (1972),  58,  1017;  W.  Meyer  Theoret.  Chim.  Acta   (1974),   35,   277;  W.  Meyer,   P.  Rosmus   J.   Chem.  Phys.   (1975),   63,   2356;  R.  Ahlrichs,   F.  Driessler,  H.   Lischka,  V.  Staemmler,  W.  Kutzelnigg  J.  Chem.  Phys.  (1975),  62,  1235;  R.  Ahlrichs  Comp.  Phys.  Comm.  (1979),  17,  31;  For  a  lucid  introduction  into  the  field  of  electron  correlation  see  Hurley,  A.C.  Electron  Correlation  in  Small  Molecules,  Academic  press,  London,  1976  24  For  example  CEPA  methods  are  not  “unitarily  invariant”,  that  is,  they  yield  different  results  if  the  occupied  or  virtual  orbitals  are  unitarily  mixed  among  themselves  which  should  not  change  the  outcome  of  a  calculation.  

so-­‐called  “quadratic  CI”  (QCISD)  method.25  It  is  very  successful  but  also  not  

variational.  While  it  is  more  complex  than  the  CISD  method,  in  practice  one  can  

apply  it  in  every  situation  where  a  CISD  or  CEPA  calculation  is  feasible.  However,  

QCISD  can  also  be  thought  of  as  an  approximation  to  an  even  more  general  and  

powerful  method:  the  so-­‐called  “coupled-­‐cluster”  method26  with  single-­‐  and  

double-­‐excitations  (CCSD).  The  CC  hierarchy  is  based  on  a  completely  different27  

Ansatz  for  the  correlated  wavefunction  that  is  size-­‐consistent  from  the  start.28  It  

leads,  however,  to  a  very  complex  set  of  equations.  With  proper  programming,  CCSD  

takes  a  limited  additional  time  over  QCISD  and  since  it  is  more  rigorous  some  

workers  prefer  CCSD  over  QCISD.  However,  the  general  accuracy  of  both  methods  is  

very  similar.      

If  a  perturbative  correction  for  the  triple-­‐excitations  is  added  to  these  methods  (the  

CCSD(T)  and  QCISD(T)  methods)  one  comes  to  the  approaches  that  are  known  to  be  

the  most  accurate  practical  single  reference  correlation  methods.  They  typically  

yield  results  of  chemical  accuracy  under  two  conditions:  (a)  The  HF  determinant  

provides  a  valid  starting  point  and  (b)  large  basis  sets  are  used.29    

2.1.3 Multireference  ab  initio  Methods  In  order  to  arrive  at  consistent  potential  energy  curves  it  is  necessary  to  calculate  a  

number  of  electronic  states  over  large  areas  of  the  potential  energy  surface  and  to  

describe  these  states  in  a  balanced  manner.  There  are  quite  a  number  of  

complications  that  theory  has  to  face  in  accomplishing  this  goal.  First  of  all  the  

character  of  a  state  may  change  very  quickly  in  certain  areas  of  the  PES.  Secondly,  

the  states  may  be  orbitally  degenerate  or  nearly  degenerate  and  this  needs  to  be  

adequately  taken  into  account  in  the  electronic  structure  calculations.  Third,  the  

states  are  also,  in  general,  of  variable  multiplicity  and  consequently,  a  balanced  

                                                                                                               25  Pople,   J.A.;   Head-­‐Gordon,  M.;   Raghavachari,   K.   J.   Chem.   Phys.   (1987),   87,   5968.   This   paper   also   introduced   the  perturbative  triples  correction  that  is  widely  used  in  conjunction  with  the  CCSD  method  26  The  literature  on  CC  methods  is  large  and  complicated.  The  closest  to  an  elementary  introduction  can  be  found  in  Hurley’s  book  and  in  a  more  recent  review:  Crawford,  T.D.;  Schaefer,  H.F.  in:  Reviews  in  Computational  Chemistry.  Vol  14,  K.B.  Lipkowitz,  D.B.  Boyd  (Eds),  Wiley-­‐VCH,  New-­‐York,  2000,  pp  33-­‐136.  27  The  Ansatz  is  exponential  rather  than  linear  as  in  the  CI  method.  Details  will  not  be  provided  in  this  course.  28  But  CC  methods  can  not  be  made  variational,  at  least  not  in  a  straightforward  way  29  The  (slow)  convergence  of  the  correlation  energy  with  basis  set  size  has  been  studied  in  chapter  1  

treatment  of  states  with  different  numbers  of  unpaired  electrons  is  necessary.  

Fourth,  states  that  cross  each  other  or  have  an  avoided  crossing  pose  particular  

challenges.  Finally,  the  electronic  structure  methods  used  must  be  able  to  correctly  

describe  the  dissociative  region,  that  is,  they  must  be  able  to  correctly  track  the  

PES’s  all  the  way  from  the  bonding  region  to  the  multiplets  of  the  isolated  atoms.    

Presently,  there  is  only  one  line  of  approach  known  that  allows  one  to  achieve  all  of  

these  objectives  simultaneously.  This  is  provided  by  so-­‐called  multireference  

methods.  The  basic  idea  is  the  following:  The  calculation  of  accurate  electronic  

energies  is  divided  into  two  steps  –  in  the  first  step,  0th  order  approximations  to  the  

states  of  interest  are  constructed  that  are  qualitatively  correct.  Thus,  there  is  a  

limited  set  of  electronic  configuration  state  functions  (CSFs)30  that  is  necessary  to  

describe  the  physics  of  the  system  correctly.  That  is,  this  set  of  CSFs  contains  all  the  

configurations  that  are  necessary  to  construct  all  target  states  with  their  correct  

multiplicity,  spatial  symmetry  as  well  as  all  those  electronic  configurations  that  are  

necessary  to  properly  dissociate  the  molecule  into  its  fragments  (atoms).  Given  a  set  

of  suitable  CSFs,  the  energy  of  the  many  electron  wavefunction  for  each  state  is  

minimized:  

 

E I( );0 =! I( );0 | H

BO|! I( );0

! I( );0 |! I( );0              

  (37)  

Here   ! I( );0  is  the  0th  order  wavefunction  for  the  I’th  electronic  state  and   HBO

 is  the  

many-­‐electron  Born-­‐Oppenheimer  Hamilton  operator  and  the  superscript  ‘0’  

reminds  us  that  we  are  only  trying  to  generate  an  energy  and  wavefunction  that  is  

qualitatively  correct  (e.g.  to  ‘0th  order’).  As  discussed  above,  we  write   ! I( );0  as  a  

linear  combination  of  the  ‚qualitative  set’  of  CSFs   !{ } :  

                                                                                                               30  A   configuration   state   function   consists   of   essentially   two   parts:   First,   the   electronic   configuration,   i.e.   the  distribution   of   electrons   among   the   available   orbitals,   and   secondly   a   spin-­‐coupling   of   the   unpaired   electrons   to  produce   the   desired   total   spin.   The   spin   coupling   is   perhaps   best   pictured   by   a   spin-­‐branching   diagram   as   for  described   elaborately   for   example   by   Pauncz   (Pauncz,   R.   Spin   Eigenfunctions.   Construction   and   use.   Plenum  Press.  New  York  and  London,  1979)  

  ! I( );0 = C

JI"

JJ#                

  (38)  

It  is  important  to  recall  that  the  CSFs   !{ }  are  themselves  built  from  a  set  of  one-­‐

electron  orbitals   !{ }  that  in  turn  are  taken  as  a  linear  combination  of  one-­‐electron  

basis  functions   !{ } .  Thus,  let  us  assume  for  simplicity  that  each  

!

J  is  a  single  

Slater  determinant;  then  one  could  write  them  as:  

  !

J= !

1

J( )....!N

J( )                

  (39)  

Where   !i

J( )  is  the  i’th  orbital  occurring  in  the  N-­‐electron  Slater  determinant  for  CSF  

J.  The  orbitals  are:  

  !

i= c

µi"

µµ!                

  (40)  

The  energy  in  eq  (37)  is  now  a  function  of  two  sets  of  wavefunction  parameters:  the  

CSF  coefficients  C  and  the  MO  coefficients  c.  Thus  we  can  write   E I( );0 = E I( );0 C,c( ) .  

The  minimization  of  this  energy  with  respect  to  both  C  and  c  leads  to  the  so-­‐called  

“Multiconfiguration  Self-­‐Consistent  Field”  (MCSCF)  method.  This  method  is  fully  

variational.31    

We  have  thus  far  been  deliberately  vague  on  the  subject  which  set  of   !{ }  is  

„suitable“  to  provide  the  desired  0th  order  description.  There  is  no  general  best  

answer  to  this  question  that  is  universally  agreed  upon.  This  then  constitutes  a  

strength  and  a  weakness  of  the  MCSCF  method:  the  choice  of  expansion  functions  

!{ }  depends  on  the  insight  that  the  user  of  the  method  has  into  the  nature  of  the  

problem  and  is  therefore  subjective.  Since  one  is  limited  in  the  number  of  0th  order  

                                                                                                               31  The  minimization  techniques  are  somewhat  involved  and  will  not  be  discussed  here.  

CSFs  for  computational  reasons,  it  is  not  uncommon  that  several  sets  are  tried  

before  the  “best”  one  is  found.  If  calculations  on  small  molecules  are  performed  a  

“safe”  choice  is  to  include  all  valence  orbitals  and  electrons  in  the  active  space  as  

long  as  this  is  possible.  Technical  limitations  restrict  the  maximum  size  of  the  active  

space  to  about  14  orbitals.  If  the  system  of  interest  has  more  valence  orbitals,  a  

choice  of  the  “important”  orbitals  for  the  problem  at  hand  must  be  made.32    

A  general  and  systematic  way  to  choose  a  set  of   !{ } ’s  is  the  following:  (1)  divide  the  

set  of  MOs  into  three  subsets:  (a)  the  “inactive”  orbitals  that  are  doubly  occupied  in  

all   !{ } ,  (b)  the  „active“  orbitals  that  are  partly  occupied  in  the  

!{ } ’s.  These  are  the  

important  orbitals  the  occupation  and  shape  of  which  changes  during  the  chemical  

reaction,  spectroscopic  transition,…  (c)  the  “virtual”  orbitals  that  are  empty  in  all  

!{ } ’s.  This  description  implies  that  the  number  of  electrons  and  orbitals  in  each  subspace  is  fixed.  Since  we  know  the  total  number  of  electrons  (N),  we  only  need  to  

specify  how  many  of  them  are  active  (n)  and  in  how  many  orbitals  (m).  (2)  In  order  

to  avoid  any  ambiguity  in  the  choice  of  CSFs,  perform  a  full-­‐CI  with  the  n-­‐electrons  in  

m-­‐orbitals.  Thus,  one  constructs  all   !{ } ’s  with  n-­‐electrons  in  m-­‐orbitals  that  are  

compatible  with  the  requested  multiplicity  and  spatial  symmetry  and  uses  this  set  

as  trial  set  fort  he  MCSCF  procedure.  Since  the  set  of   !{ }  is  complete  in  this  many-­‐

electron  subspace  one  speaks  of  the  „complete  active  space  self-­‐consistent  field”  

(CASSCF(n,m))  method.33    

One  subtlety  is  the  following:  the  optimal  MO  coefficients  c  are  different  for  different  

states.  It  may  therefore  become  very  laborious  to  optimize  a  new  set  of  c’s  for  each  

                                                                                                               32  It  may  not  always  be  possible  to   include  all  of  the  valence  orbitals   for  another  reason:   lack  of  convergence!   If  an  active  orbital  becomes  almost  doubly  occupied  or  almost  empty,  convergence  problems  will  result  in  the  CASSCF.  In  this  case  it  will  be  advantageous  to  not  include  these  orbitals  into  the  active  space.  33  For   CASSCF   and  MRPT  methods   (CASPT2)   see   K.   Andersson,   B.   O.   Roos,   in:   D.   Yarkony   (Ed.),  Modern   Electronic  Structure  Theory.  World  Scientific,  Singapore,  1995,  p.  55  

state  of  interest.34  Consequently,  one  tries  to  determine  a  single  set  of  c’s  that  

represent  the  best  compromise  for  all  states  of  interest.  Thus,  one  writes:  

    ESA!CASSCF = w

IE I( );0

I"              

  (41)  

With  constant,  user-­‐supplied,  weights   wI  that  sum  to  unity.  Minimization  of  this  

energy  expression  results  in  a  set  of  “democratic”  orbitals  for  the  states  of  interest  

as  well  as  0th  order  approximations  for  all  states  of  interest.    

In  quantum  chemical  language,  the  correct  0th  order  description  that  was  thus  

generated  takes  care  of  all  static  (near-­‐degeneracy)  correlation  effects.  What  is  left  to  

be  done  is  to  try  to  recover  as  much  as  possible  of  the  remaining  (dynamic)  

correlation  energy.  This  can  be  approached  either  by  perturbation  theory  (leading  

to  “multireference  perturbation  theory”  MRPT  methods)  or  by  configuration  

interaction  techniques  (leading  to  “multireference  configuration  interaction”,  

MRCI).35  What  is  done  in  these  techniques  is  to  regard  either  the  entire   ! I( );0  or  its  

constituent  parts   !

J  as  „reference“  wavefunctions  and  to  perform  single-­‐  and  

double-­‐excitations  relative  to  these  functions.  This  produces  a  set  of  “excited”  CSFs  

!

J  which  are  used  as  expansion  sets  of  the  many-­‐electron  problem.  Thus  one  

writes:  

  ! I( ) = B

JI"

JJ#                

  (42)  

With  parameters   BJI  that  are  determined  by  variational  techniques  (MRCI)  or  

perturbational  techniques  (MRPT).  Since  the  set  of   !

J  may  include  the  set  of  

!

J,  

                                                                                                               34  In   addition   one  would   have   to   calculate   transition   properties   from   sets   of   non-­‐orthogonal   orbitals   used   for   the  expansions  of  the  states  involved  in  the  transition.  This  is  possible  but  much  more  laborious  than  calculations  based  on  a  single  orthonormal  set  of  orbitals.  35  The  “ultimate  goal“  of  a  consistent  and  efficient  „multireference  coupled  cluster“  (MRCC)  theory  has  so   far  been  plagued  by  massive   technical   problems  and   it   is   presently  not   clear  whether   such  methods   can  be  developed   into  large-­‐scale  research  tools  for  non-­‐experts.  

we  do  not  write:   ! I( ) = ! I( );0 + B

JI"

JJ#  -­‐  in  general,  the  coefficients  C  of  those  

CSFs   !

J  that  are  contained  in  the  

!

J  set  are  “revised”  by  the  dynamic  

correlation  treatment.36  The  set  of   !

J’s  may  be  very  large  and  may  have  hundreds  

of  millions  of  members.  While  such  large  scale  calculations  are  barely  feasible  using  

todays  most  advanced  programs  and  computers  it  is  highly  desirable  to  come  to  as  

compact  sets  of  CSFs  as  possible  and  this  is  still,  even  after  decades  of  research,  a  

topic  for  method  development.  We  will  not  go  into  further  detail  here,  but  simply  

note  that  the  ORCA  program  is  of  the  “individually  selecting”  type  in  its  MRCI  and  

MRPT  modules  –  a  technique  that  was  pioneered  and  extensively  practised  in  Bonn  

for  a  long  time  and  with  major  success.37  Thus,  the  program  contains  two  sets  of  cut-­‐

offs:  

• A   parameter   Tsel   (typically   10-­‐5-­‐10-­‐7   Eh)   that   determines   which   !

J  are  

important   an   thus   are   “allowed”   to   enter   the   variational   space   in   a   MRCI  

calculation.  These  are  the  ones  that  interact  most  strongly  with  the  reference  

wavefunction   ! I( );0  and,   in   practice,   are   selected   on   the   basis   of   second-­‐

order   perturbation   theory.   This   selection   procedure   selects   typically  ≈104-­‐

106  CSFs  for  the  variational  treatment.  The  remaining  (much  larger)  number  

of  CSFs  are  simply  treated  by  second-­‐order  perturbation  theory.  

• A   second   parameter   Tpre   (typically   10-­‐2-­‐10-­‐5)   determines   which   !

J  are  

allowed   to   act   as   reference   configurations   in   a  MRCI   or  MRPT   calculation.  

                                                                                                               36  This   “revision”  may   be   quite   important   since   for   a   number   of   problems   the   CASSCF  method   does   not   produce  accurate   initial   values.  An   important  case   is  met   in   the  calculation  of  exchange  coupling  constants  as  discussed   for  example  by  Malrieu  and  co-­‐workers   (Miralles,   J.;  Castell,  O.;  Caballol,  R.;  Malrieu,   J.  P.  Chem.  Phys.  1993,  102,  103;  Miralles,  J.;  Daudey,  J.  P.;  Caballol,  R.  Chem.  Phys.  Lett.  1992,  198,  555;  Castoll,  O.;  Miralles,  J.;  Caballol,  R.  Chem.  Phys.  1994,  179,  377)  and  Staemmler  and  co-­‐workers.(Fink,  K.;  Fink,  R.;  Staemmler,  V.  Inorg.  Chem.  1994,  33,  6219;  Fink,  K.;  Wang,  C.;  Staemmler,  V.  Int.  J.  Quant.  Chem  1997,  65,  633;  Fink,  K.;  Wang,  C.;  Staemmler,  V.  Inorg.  Chem.  1999,  38,  3847)  37  R.  J.  Buenker,  S.  D.  Peyerimhoff  Theoret.  Chim.  Acta  35  (1974)  33;  R.  J.  Buenker,  S.  D.  Peyerimhoff  Theoret.  Chim.  Acta  39  (1975)  217  

Each   !

J  that  contributes  with  a  weight  of  at  least  Tpre  to  any  of  the  

! I( );0  

will  be  part  of  this  “privileged”  set  of  CSFs.    

The  final  subject  to  be  mentioned  is  a  fundamental  weakness  of  MRCI  –  its  lack  of  

size  consistency.  That  is,  calculated  energies  of  noninteracting  subsystems  do  not  

exactly  sum  to  the  same  energy  as  the  same  system  calculated  as  a  “supersystem”.38  

As  discussed  in  detail  in  the  lectures  on  quantum  chemistry,  this  weakness  is  shared  

by  all  CI  based  approaches  and  is  related  to  the  neglect  of  higher  than  doubles  

excitations.  A  straightforward,  approximate  and  essentially  empirical  remedy  for  

this  problem  is  the  so-­‐called  “multireference  Davidson  correction”39  which  is  

applied  to  the  MRCI  energies  to  yield  the  so-­‐called  “MRCI+Q”  method  which  will  be  

used  in  this  experiment.  

2.2  Description  of  the  Experiment  In  the  actual  experiment,  we  will  study  the  CH,  NH  and  OH  molecules  (five  valence  

orbitals  and  5-­‐7  valence  electrons)  that  will  be  correlated  in  this  experiment  with  

multireference  methods.  We  will  also  treat  F2  and  N2  using  single  reference  methods  

(14  and  10  valence  electrons  and  10  valence  orbitals  respectively).  The  

experimental  data  for  these  molecules  is  reproduced  in  Table  1.  Table  1:  Experimental  data  for  some  low-­‐lying  states  of  some  diatomic  molecules.  

Molecule   State   Te  (cm-­‐1)   ωe  (cm-­‐1)   ωexe    (cm-­‐1)   Re  ( Å)  CHa   2Σ+   31801   2840   126   1.1143     2Σ-­‐   (26044)   (1795)   -­‐   (1.1975)     2Δ   23189   2931   97   1.1019     4Σ-­‐   (5844)   (3145)   (72)   (1.085)     2Π   0   2859   63   1.1199  NHb   1Π   (43744)   (2122)   -­‐   1.1106     3Π   29807   3231   99   1.0369     3Σ+   21202   3352   74   1.0360     1Δ   (12566)   3188   (68)   1.0341     3Σ-­‐   0   3282   78   1.0362  OHc   2Σ+   32684   3179   93   1.0121  

                                                                                                               38  In  multireference  CI  calculations  the  problems  are  not  overwhelming  if  the  reference  space  is  chosen  wisely.  Single  reference  CI  calculations  suffer  severely  from  the  problem.  39  G.  Hirsch,  P.  J.  Bruna,  S.  D.  Peyerimhoff,  R.  J.  Buenker  Chem.  Phys.  Lett.  52  (1977)  442  

  2Π   0   3738   85   0.9697  HFd   1Σ+   0   4138   90   0.9171  F2e   1Σg+   0   917   11   1.4119  N2f   1Σg+   0   2358   14   1.0977  a  –  Ionization  potential  =  10.64  eV;  D0=  3.46  eV  b  –  Ionization  potential  =  (13.36)  eV  D0 ≤  3.47  eV  c  –  Ionization  potential  =  12.9  eV,  D0=  4.39  eV  d  –  Ionization  potential  =  16.06  eV,  D0=  ≤6.4  eV  e  –  Ionization  potential  =  15.686  eV,  D0=  1.602  eV  f  –  Ionization  potential  =  15.580  eV,  D0=  9.759  eV    

2.2.1 Single  Reference  Calculations  In  this  section,  we  will  compare  some  single  reference  calculations  amongst  each  

other  and  with  experiment.  In  order  to  provide  a  first  orientation,  use  a  small  SVP  

basis  set  and  try  to  calculate  the  entire  potential  energy  surface  up  to  the  

dissociation  limit.  We  have  deliberately  chosen  a  molecule  with  a  single  bond  (F2)  as  

well  as  a  molecule  with  a  triple  bond  (N2)  in  order  to  demonstrate  the  possibilities  

and  limitations  of  the  methods  used.  We  will  compare  the  RHF,  UHF,  MP2,  QCISD(T)  

and  CCSD(T)  methods.    

• Calculate  a  PES  for  the  dissociation  process  with  the  RHF  method  using  the  

input  file  below;  repeat  in  a  similar  way  for  F2  and  N2.  What  do  you  observe?  

Is  the  result  physically  acceptable?  Calculate  the  isolated  F  and  N  atoms  using  

the  UHF  method   and   compare  with   your  dissociation   limit   in   as   far   as   you  

can  define  one!    

 The  result  of  this  calculation  will  be  a  table  of  numbers  that  list  the  value  of  the  

parameter  that  you  have  scanned  and  the  energy  that  you  have  calculated.  This  

# # Potential energy surface of HF with RHF # ! RHF SVP TightSCF # This line define a parameter R that is to be varied from 1.2 to 3.5 # Angström in a total of 35 equidistant steps. Other definitions of PES # are of course possible (please refer to the ORCA manual) %paras R= 0.85,3.5,40 end * xyz 0 1 F 0 0 0 H 0 0 {R} *

appears  in  two  places  in  the  ORCA  output:  (a)  in  the  output  file  itself,  and  (b)  to  

make  you  life  easy,  there  will  also  be  a  file  written  that  is  called  “*.trjscf.dat”.  

“TRJ”  stands  for  “trajectory”  and  “SCF”  for  SCF  energy.  If  you  do  other  calculations  as  

well  there  will  be  additional  files  “*.trjxxx.dat”  where  xxx  may  be  “mdci”  (for  

single  reference  correlation  methods)  “mp2”  (for  the  MP2  method),  “mrci”  (for  

calculations  with  the  MRCI  module)  “casscf”  (for  individual  states  in  a  state-­‐

averaged  CASSCF  calculation)  or  “cis”  (for  CIS  or  TD-­‐DFT  calculations).  These  files  

can  be  directly  imported  into  plotting  or  data  analysis  programs.    

As  you  will  find  in  your  calculations  above  that  the  RHF  method  does  not  lead  to  

potential  curves  which  dissociate  correctly,  it  is  possible  to  enforce  this  on  the  

method  using  a  special  type  of  UHF  wavefunction,  namely  one  of  “broken  

symmetry”.  In  this  case,  the  spin-­‐up  and  spin-­‐down  orbitals  of  the  UHF  solution  are  

forced  to  localize  on  one  of  the  atoms.  The  solution  has  the  correct  physics  and  

energy  but  an  erroneous  spin  density.  It  is,  nevertheless,  a  feasible  way  to  simulate  

some  static  correlation  effects  with  single  determinant  methods.40  The  input  for  this  

method  is  shown  below.  

 JOB  

• Plot  the  two  PESs  and  compare  them.  Determine  the  dissociation  energy  for  

the  UHF  solution.    

                                                                                                               40  In  inorganic  chemistry  this  is  a  quite  important  subject  in  the  framework  of  exchange  coupling  constant  prediction  based  on  DFT  methods.  See  chapter  Error!  Reference  source  not  found.  on  page  226.  

# # Potential energy surface of HF with UHF # ! UHF SVP TightSCF # Now we scan the opposite direction since we need to find the correct # solution for the dissociation limit before “moving in” %paras R= 3.5,0.8,40 end # This instructs the program to flip the spin on the second fluorine # following an initial spin triplet calculation. We have a single bond, # therefore we look for a broken symmetry solution with one localized # electron per site; in the N2 case it is three electrons. Brokensym 1,1 * xyz 0 1 F 0 0 0 H 0 0 {R} *

Now  that  we  have  a  HF  solution,  let  us  try  to  do  the  correlated  calculations.  At  the  

moment  this  is  only  feasible  with  the  ORCA  program  based  on  a  RHF  solution.    

JOB:  

• Re-­‐run  the  RHF  calculation  but  include  the  keywords  (in  turn):  QCISD(T),

CCSD(T)  and  CISD.    

• Repeat  the  calculations  for  N2  and  F2.  

Compare  your  findings  –  do  the  correlated  methods  dissociate  correctly?  If  not,  

what  is  your  interpretation?    

Now  that  the  applicability  of  different  methods  is  understood  let  us  turn  to  a  more  

serious  calculation  and  examine  the  vicinity  of  the  minimum  in  order  to  predict  

spectroscopic  constants.  From  your  initial  calculations  you  have  a  fairly  good  idea  

where  the  minimum  occurs  on  the  PES  of  the  two  molecules  with  which  method  

(see  also  Table  1).  Now,  do  a  PES  scan  around  the  minimum  with  a  large  basis  set  in  

order  to  obtain  accurate  results.  On  the  order  of  10  steps  should  be  sufficient  to  give  

a  reliable  fit.41  

 JOB:  

                                                                                                               41  In   fact,   if  you  are  serious  about  science  you  always  CHECK  on  everything  you  do  as  carefully  as  possible   -­‐  This   is  independent  of  whether  one  does  experiments  or  performs  calculations!   In   the  present  case   this  would  mean  that  you  check  (with  a  small  basis  set  and  perhaps  with  the  HF  method)  how  many  points  you  need  and  how  far  a  region  around   the  minimum   you   have   to   study   in   order   to   get   stable   results.   If   you   choose   a   region   roughly  ±0.1-­‐0.2   Å  around  the  minimum  and  have  spacings  between  your  points  of  roughly  0.01-­‐0.02  Å  the  results  should  be  reasonably  converged.  If  you  choose  the  region  too  small  you  can  not  calculate  reliable  anharmonic  constants  since  you  are  well  in  the  quadratic  region;  if  you  choose  the  spacing  too  large,  the  region  around  the  minimum  is  not  well  described  by  a  Morse  potential.    

# # Potential energy surface of HF with correlation methods # ! RHF TZVPP QCISD(T) TightSCF Conv # %paras R= 0.81,1.01,12 end * xyz 0 1 F 0 0 0 H 0 0 {R} *

• Do  the  calculations  with  RHF,  MP2,  QCISD,  CCSD,  QCISD(T),  CCSD(T)  and  the  

B3LYP-­‐DFT   method   for   HF,   F2   and   N2.   To   include   MP2   is   instructive   for  

comparing  higher  and  lower  level  correlation  methods.  

• Compare   first   the   MP2   with   the   CCSD   results.   How   important   are   higher  

order   correlation   effects   brought   in   by   the   iterative   nature   of   the   CCSD  

method?  

• Compare   the   CCSD   with   the   CCSD(T)   results.   How   important   are   the  

connected  triple  excitations?  

• For   the   CCSD(T)   method,   study   the   basis   set   dependence   by   choosing   a  

smaller  basis  set  (SVP),  a  basis  set  lacking  the  f-­‐polarization  set  (TZV(2d,2p)),  

a   basis   set  with   only   one   set   of   polarization   functions   (TZVP)   and   finally   a  

very  large  basis  set  (QZVP).  Since  you  get  the  SCF  results  automatically  too  –  

what   is   your   conclusion   about   the   basis   set   dependence   of   correlated  

methods?  What   types  of   correlation  effects  are  you  describing  by  enlarging  

the  basis?    

• Determine  the  spectroscopic  constants  and  compare  them  with  experiment.  

Produce   tables   with   your   results?   Are   the   ab   initio   calculations   reliable?  

Which  method  shows  the  best  agreement  with  experiment?42    

The  latter  fit  you  can  either  do  yourself  with  xmgrace  (chapter  Error!  Reference  

source  not  found.  on  page  Error!  Bookmark  not  defined.).  Alternatively,  there  is  

a  small  fitting  program  in  the  ORCA  package  that  can  do  the  job  (orca_fitpes)  

and  that  is  called  automatically  at  the  end  of  a  surface  scan  for  a  diatomic  

molecule.43    

                                                                                                               42  In   fact,   in   the   case   of   HF   you   can   walk   in   the   footsteps   of   the   great   pioneers   of   accurate   calculations:  Meyer,  Ahlrichs,  Kutzelnigg  and  Staemmler  who  developed  the  CEPA  method  in  the  early  1970’s  and  applied  them  with  great  success  and  impressive  results  to  small  molecules.  To  do  CEPA  calculations  best  choose  CEPA/1  instead  of  CCSD(T)  in  the  input  line.  See  CEPA  references  given  on  page  174.  43  You   can   also   call   this   module   in   a   stand-­‐alone   mode.   To   this   end   you   have   to   edit   the   file  *.trjxxx.dat”  and  add  a  some  information  on  the  first  line:  1 11 1 14 14 <<<<---- additional required information

2.2.2 Multireference  Calculations  We  will  now  turn  to  multireference  methods  which  can  be  applied  to  the  ground-­‐  

and  excited  states  over  the  entire  PES.  In  general  we  scan  the  PES  from  “out-­‐to-­‐in”  in  

order  to  help  the  program  to  initially  select  the  correct  set  of  orbitals  that  must  be  

correlated.  We  start  by  a  SA-­‐CASSCF  calculation  and  include  the  states  listed  in  

Table  1.  It  is  a  good  idea  to  start  with  a  small  basis  set  and  only  do  the  final  

calculations  with  the  largest  (target)  basis  set.  A  suitable  ORCA  input  for  the  CASSCF  

calculations  is:  

 

 JOB:  

• Plot  the  result  contained  in  the  “*.trjcasscf.dat”  file.    

Now  we  scan  the  region  around  the  minima  using  the  more  accurate  MRCI+Q  

method.  In  order  to  do  this,  you  have  to  add  a  block  like:  

                                                                                                                                                                                                                                                                                                                                          1.20000000 -108.80922141 <<<<---- ORCA output

1.18000000 -108.82097001

1.16000000 -108.83134251

Etc

The  new  title  contains  the  following  items: NPES NPTS ISANG MA MB.  Where:  NPES –    Number  of   potential   energy   surfaces   (columns  of   data)  NPTS –    Number  of   points   (rows  with  data)  ISANG –     Set   to  unity   if   the  distance   is   in  Angström,  otherwise  units  of  Bohr   are   assumed  MA,MB – Masses  of  the  two  nuclei  (in  units  of  the  proton  mass,  e.g.  C=12).  This  new  file  is  run  through  orca_fitpes   in  order  to  determine  the  various  spectroscopic  constants  which  can  then  be  compared  to  the  experimental  data.  (orca_fitpes myjob.trjmdci.dat >myjob.trjmdci.out)  

! RHF SVP TightSCF Conv %casscf nel 5 # number of active electrons Norb 5 # number of active orbitals mult 2,4 # multiplicities doublet and quartet nroots 6,1 # 6 doublet roots, 1 quartet root switchstep nr # second order converger ON end %paras R= 3.5,1.0,40 end # scan all the way inwards * xyz 0 2 C 0 0 0 H 0 0 {R} *

 JOB:  

• Use   the   orca_fitpes   program   to   analyze   the   results   of   the   CASSCF  

calculations 44  (in   the   “*.trjcasscf.dat”   files)   and   of   the   MRCI  

calculations  (in  the  “*.trjmrci.dat”  files).    

• Compare  your  results  with  the  experimental  data  in  Table  1.  Document  your  

mean  error,  maximum  error  and  mean  absolute  error  relative  to  experiment.    

• Increase  the  basis  set  to  TZVPP.  Do  your  results  get  better?    

• Vary  the  values  of  the  parameters  Tsel  and  Tpre.  How  do  your  results  react  to  

these  changes?    

ADDITIONAL  CALCULATIONS  I  (VOLUNTARY)  

• Use  eq  (29)  to  calculate  the  Franck-­‐Condon  factors  for  transition  between  the  

lowest   ground   state   vibrational   levels   and   vibrational   levels   of   the  

electronically  excited  states  in  the  harmonic  approximation.    

 

• Plot  the  intensity  distribution  in  the  vibrational  progression.  If  you  “broaden  

it  out”  by  convoluting  the  calculated  transitions  with  Gaussian  curves  -­‐  where  

is  the  band  maximum?  How  much  does  it  deviate  from  the  0-­‐0  transition  and  

                                                                                                               44  If  you  have  crossings  of  levels  in  your  PES  you  have  to  be  careful.  In  the  case  of  discontinuities  the  Morse-­‐fits  fail.  Thus,  you  have  to  make  sure  that  your  columns  contain  the  correct  data  that  correctly  tracks  the  state  of  interest.  You  have  to  look  at  the  results  of  the  Morse  fits.  Sometimes  they  converge  to  nonsensical  values.    

%mrci ewin -3, 1e10 citype mrci tsel 1e-6 # Selection parameter tpre 1e-4 # pre-selection parameter intmode fulltrafo newblock 2 * # calculate doublet states nroots 6 # six roots (as in CASSCF) refs cas(5,5) end # the reference space end newblock 4 * # calculate quartet states nroots 1 # one root (as in CASSCF) refs cas(5,5) end # SAME reference space! end end

the   vertical   transition   energy?  What   does   this   implicate   for   comparison   of  

calculated   vertical   transition   energies   and   band   maxima   for   larger  

molecules?  

ADDITIONAL    CALCULATIONS  II  (VOLUNTARY)  

• Compare   single-­‐   and  multireference  methods   for  N2.  To   this   end  perform  a  

CASSCF(6,6)45  calculation  on  N2     and  perform  a  MRCI+Q  PES  scan  as   in   the  

first  part  of  the  experiment.  

• Compare  CASSCF  with  RHF  

• Compare  MRCI+Q  with  QCISD  and  CEPA/2.  

 

ADDITIONAL  CALCULATIONS  III  (VOLUNTARY):  

• How   would   you   calculate   the   ionization   potential   of   the   molecules   using  

multireference  methods?    

• How  would  you  consistently  calculate  the  dissociation  energies  De´and  D0?    

                                                                                                               45  This   is  not  perfectly   straightforward   since   close   to   the  equilibrium  distance   the  program  may  have  difficulties   to  “find”  the  high-­‐lying  σ*  orbital  which  is  required  to  complete  the  2p-­‐valence  space.  To  this  end,  start  the  calculation  from  the  natural  orbitals  of  a  MP2  calculation  (a  commonly  used  trick  to  start  CASSCF  calculations):    ! RHF MP2 SVP TightSCF

%mp2 density relaxed natorbs true end

%paras R= 1.20 end

* xyz 0 1

N 0 0 0

N 0 0 {R}

*

$new_job

! SVP TightSCF moread

%moinp "N2-CASSCF-01.mp2nat"

%casscf nel 6 norb 6 switchstep nr end

%paras R= 1.20,1.0,12 end

* xyz 0 1

N 0 0 0

N 0 0 {R}

*  

• Do  these  calculations  and  compare  your  results  with  the  experimental  data  in  

Table  1.  Also,  compare  the  D-­‐values  that  you  get  with  the  ones  obtained  from  

the  Morse  fits.  Are  the  Morse-­‐fit  values  reliable?  

ADDITIONAL  CALCULATIONS  IV  (VOLUNTARY):  

• The  only  of  these  molecules  where  DFT  is  expected  to  provide  a  reasonable  

description  is  NH.    

• Do  a  DFT/TD-­‐DFT  calculations  on  the  ground-­‐  and  excited  states  of  NH  and  

compare  your  results  with  the  ab  initio  calculations.  How  accurate  is  the  DFT  

methodology?