Transcript Slide 1

Helsinki Finland
June 22-27 2009
Chair: Manuel Yanez
15:20 Wednesday June 24 2009
New Variational Approaches to Excited and
(Nearly) Degenerate States in Density
Functional Theory
Tom Ziegler Department of Chemistry
University of Calgary
Calgary Alberta Canada
Excited States
DFT-methods
1: DSCF (Slater, 1972)
Excited States
2: TD-DFT (Gross,Cassida 1995)
I.
Demonstrate that the basic
equation
of TD-DFT can be derived from a timeindependent constraint variational
DFT (CV-DFT) procedure
Ground state
II. Discuss the the relation between
DSCF and TD-DFT as special cass of
CV-DFT
III. Application to charge transfer
IV. Dissociation of molecules
Basic Ground State KS-theory
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102
A closed shell molecule is described in the Kohn-Sham formulation
of density functional theory (KS-DFT) by a single Slater determinant:
0  123 .....i j .....n
with the corresponding IDEMPOTENT density matrix given by:
occ
(1,1')  i* (1')i (1)
i
The density matrix is optimized in such a way that it minimizes the energy expression

E KS  ET VNe  E H  E XC,KS
Basic Ground State KS-theory
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102
The density matrix is optimized in such a way that it minimizes the energy expression
E KS  ET VNe  E H  E XC,KS
Here the kinetic- and nuclear attraction energy is given by
ET VNe 



hˆ 0 (1') (1,1')
(11')
d1
The Hartree Coulomb interaction energy by
1
EH 
2

1
 (1,1) r (2,2)d1d 2
12
Exchange correlation energy

E XC,KS  E XC,KS [(1,1)]
Basic Ground State KS-theory
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102
The set of spin-orbitals that optimize the ground state Slater determinant
can be found as solutions to the one-electron KS-equation
Where
With the exchange correlation potential VXC given by
Electronic Ground-State Hessian
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102
As a first step towards a variational theory for the determination of excitation
energies by KS-DFT we consider a variation of each of the occupied spin-orbitals
Here Uai are our variational parameters
From the variations i +di we can form a set of orbitals orthonormal to second order in U
From this set we can generate a new determinantal wave function
 123.....ij .....n

Electronic Ground-State Hessian
Corresponding to the determinantal wave-function
 123.....ij .....n

We have to second order in U the density matrix
Electronic Ground-State Hessian
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102
Substituting into the expression for EKS affords to second order
KS

A
1
EKS [ 0  D ']  EKS [ 0 ] U * U  KS
2
B
BKS U  [3]
 * O [U]
KS 
A U 
Here
KS
KS
KS
KS
Aai,bj
 dijdab (a  i ) Kai,bj
; Bai,bj
 Kai,jb
Where
K
KS
ai,bj
K
C
ai,bj
K
KS,XC
ai,jb


1
*

(1)

(1)

(2)


i
b
j (2)d 1d 2
r12
*
a
  a* (1)i (1) f XC (1)b (1)*j (1)d 1
Stationary Points and Excitation Energies
We now have to second order
KS

A
1
0
0
EKS [  D ']  EKS [ ] U * U  KS
2
B
BKS U  [3]
 * O [U]
KS 
A U 
We shall next find stationary points U(I) on EKS[0+D], such that
 dE 
 0 ; (a 1,vir;i 1,occ)


dUai (U (I ) )
And such that
KS

A
1
0
0
( I)
(I)
E
[


D

']
E
[

]

U
*
U
 KS


KS
 KS
2
B
represents an excitation energy
BKS U (I ) 
 ( I )* 
KS 
A U 
Stationary Points and Excitation Energies
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102
In the expression for the new density
 (1,1')   0 (1,1')  D(1,1')
occ vir
occ vir
  0 (1,1')  U aii* (1)a (1)  U ai* a* (1)i (1)
i
a
i
occ vir vir
a
occ occ vir
*
U U bi (1)b (1)  U ak
U aii* (1)k (1)  O (3) [U]
*
ai
i
a
*
a
b
i
k
a
Only the last two terms give rise to a change in the energy if ois optimized
occ occ vir
*
   U ak
U aii* (1) k (1)
occ
occ
i
k
a
vir
occ vir vir
vir

Uai* Ubia* (1)b (1)
i
a b
Stationary Points and Excitation Energies
Only the last two terms give rise to a change in the energy if ois optimized
occ occ vir
*
Docc   U ak
U aii* (1) k (1)
occ
occ
i
k
a
vir
vir

occ vir vir
*
Dvir  Uai
Ubia* (1)b (1)
i
a b
We have for the total charge in Docc
occ occ vir
 D(1)occ d    
i
k

*
Uak
Uaii* (1) k (1)d
occ occ vir
  
a
i
k
*
Uak
Uaidik
occ
*
 Uai
Uaidik  Dqocc
a
i
Whereas the total charge in Dvir is given by
occ vir vir
occ vir vir
occ
 D(1)vir d   Uai* Ubia* (1)b (1)d  Ubi* Uaidab  Uai* Uai  Dqvir
i
b a
i
occ
We have that:
b a
*
Dqvir  Dqocc  U ai
U ai
i
i
Stationary Points and Excitation Energies
We have a transfer of charge
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102
occ
*
U
 aiUai
occ
vir
vir
occ
i
Dqvir
Dqocc

In single excitations we shall require that a single charge is transferred

occ

*
U
 aiUai  1
i
This is a generalization of DSCF-DFT where an electron is promoted from
an occupied to a virtual orbital

i
a
Stationary Points and Excitation Energies
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102
We now have that the stationary points for E’ that fulfills the condition:
 U U
i
a
*
ai
ai
1
Introducing
KS

A
1
L  EKS [  0  D '] EKS [ 0 ] U * U  KS
2
B

 (  U ai* U ai 1)
i
a
and  is a Lagrange multiplier
We demand
dL / dU ai  0
B KS U 
 * 
KS 
A U 
Stationary Points and Excitation Energies
Thus the requirement that dL/dUai leads to
We must thus have
in order for dL/dU =0 to hold for any variation dU
Stationary Points and Excitation Energies
AKS
 KS*
B
( I) 


B
( I) U
   ( I )* 
KS* 
A 
U 
KS
For non-relativistic cases with AKS and BKS real

By making use of the Tamm-Dankoff approximation BKS=0
This equation is identical to that obtained from TD-DFT/TD within
the adiabatic approximation
Derivation of transition moments
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102
In wave function mechanics we have from Raleigh-Schrödinger perturbation theory
for the polarizability tensor
rs  2
I0
Re  0 Dˆ r I I Dˆ s  0
EI  EO
Here
Electric dipole operator


0 Dr I
E I  E0
Electric transition dipole moment component
Excitation energies
rs  2
Derivation of transition moments
Re  0 Dˆ r I I Dˆ s  0
I0
EI  EO
In DFT the static polarizability tensor component rs is given by:
rs  2d [A  B ] d
(r)
KS
KS 1

(s)T
Ziegler, et al J.Chem
Phys. 2007,126,174103
Here
dai(r)  a r i
dai(s)T  i s a  a s i
Writing next the real matrix [AKS+BKS]-1 in its spectral resolution form
( I) ( I)T 

U
U
KS 1
[A KS  B
]   

I
  I 
affords

d (r)U ( I ) (d (s)U ( I) )T 
rs  2I 




I
or by comparison to the Raleigh-Schrödinger expression

(r) ( I )
 (r),KS

d
U
IO
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102

Derivation of transition moments
For a component of the electronic magnetic susceptibility tensor krs,
Raleigh-Schrödinger perturbation theory affords the following SOS expression
k rs  2
Re  0 Mˆ r I I Mˆ s  0
EI  EO
I0
Here :

In DFT we find :
Here :
k rs  m S S m
(r) 1/2 1/2
S  A

Ziegler, et al J.Chem
Phys. 2007,126,174103
ˆ s a   a m
ˆ s i
mai(s)T  i m
ˆ r i
mai(r)  a m

(s)T
KS
B

KS 1
Derivation of transition moments
k rs  2
I0
In DFT we find :
k rs  m S S m
(r) 1/2 1/2
J

We get :
EI  EO
Ziegler, et al J.Chem
Phys. 2007,126,174103

Making use of the fact
I   U ( J )U ( J )T
(s)T
Re  0 Mˆ r I I Mˆ s  0
  U ( J )U ( J )T  
( J )U ( J )U ( J )T 
(J)
(J)


 

 J
( J )
( J )



J 



(r ) (1/2) ( J )
(s ) (1/2) ( J )

m
S
U

m
S
U
J
J
k rs  2J 

J




T




or by comparison to the Raleigh-Schrödinger expression

(r),KS
mJO
 J m(r)S1/2U ( J)
Ziegler, et al. J. Chem.
Phys. 2009, 130,154102
Charge Transfer
Ziegler, et al. J. Chem. Phys. 2008, 129, 184114
Time-dependent density functional theory (TD-DFT) at the
generalized gradient (GGA) level of approximation has
shown systematic errors in the calculated excitation energies.
This is especially the case for energies representing electron
transitions between two separated regions of space or between
orbitals of different spatial extent.
1
0  i0i0  ii
3
i0
Ground state
1  i0a0  ia
a0

Triplet
excited state
21
Charge Transfer
Ziegler et al.
Theochem. 2009
1
R
0  i0i0  ii
Ground state


K
0
a
1  i0a0  ia
Triplet excited state

HF
E(3 1)  E(10 )  a0  i0  Kaa,i
i
1.
DE-HF 
HF
rs ,tq
3
0
i
HF-Case
TD-HF
Ziegler, et al. J. Chem.
Phys. 2008, 129, 184114

HF
  a0  i0  Kaa,i
i
2.

 d r s d t  q

d r t d s q
Same results
1
*

(1)

(1)

(2)


s
t
q (2)d 1d 2
r12
*
r
1
*
HF
HF ,XC

(1)

(1)

(2)

(2)d

d


K

K

t
s
q
1
2
rs ,tq
rs ,tq
r12
*
r
22
Ziegler et al.
Theochem. 2009
1
Ziegler, et al. J. Chem.
Phys. 2008, 129, 184114
Charge Transfer
R
0  i0i0  ii
Ground state

3

0
i
0
a
1  i0a0  ia
Triplet excited state

KS-Case
DE-KS 1.
TD-DFT
2.
1 KS
1 KS
KS
E(  )  E(  )      Kaaaa  Ki i i i  Kaa,i
i
2
2

3
1
1
0
0
a
0
i
  a0  i0  KaiKS,ai  KaiKS,i a
Different result

 K
KS
rs ,tq
 d r s d t q
1
*

(1)

(1)

(2)


s
t
q (2)d 1d 2
r12
*
r
d r t d s q  r* (1)t (1) f x (1)s (1)q* (1)d 1d 2  KrsKS,tq  KrsKS,XC
,tq
23
Charge Transfer in steps
Ziegler et al.
Theochem. 2009
1
R 
      ii
0
0
i
0
i
Ground state

3


0
i
0
a
1  i0a0  ia
Triplet excited state

I. Separation step
Frozen orbitals

HF-Case
KS-Case


DE elHF
DE
KS
el
J. Neugebauer, O. Gritsenko, and
E. J. Baerends, J. Chem. Phys. 124,
214102 2006 .
M. J. G. Peach, P. Bernfield, T. Helgaker,
and D. J. Tozer, J. Chem. Phys.
128, 044118 2008 .
A. Dreuw and M. Head-Gordon,
J. Am. Chem. Soc. 126, 4007 2004
24
.
Ziegler et al.
Theochem. 2009
Charge Transfer in steps
II. Ionization
Frozen orbitals
i0

HF-Case
I  dE HF / dni (0) Dni 
 
0,HF
i
1 2 HF 2
d E / d ni (0) Dni2

2
1 HF
 K i i ,i i   i0,HF
2
1 2 KS 2
I  dE / dni (0) Dni  d E / d ni (0) Dni2
2

KS-Case
1
  i(0),KS  K iKS
2 i ,i i
KS
25
Charge Transfer in steps
Ziegler et al.
Theochem. 2009
III. Electron attachment
Frozen orbitals
a0
A  dE HF / dni (0) Dna 
HF-Case 


KS-Case
0,HF
a
A  dE
1 2 HF 2
d E / d ni (0) Dna2

2
1 HF
 K aa,aa   a0,HF
2
KS
1 2 KS 2
/ dna (0) Dna  d E / d na (0) Dna2
2
1 KS
  a0,KS  K aa,aa
2
26
Charge Transfer
Ziegler et al.
Theochem. 2009
1
R
0  i0i0  ii
Ground state

3

0
i
1  i0a0  ia
Triplet excited state
0
a
IV. Recombining


HF
 HF
DE

K
el
aa,i i
HF-Case
KS-Case
DE
KS
el
K
KS
aa,i i

27
Charge Transfer
Ziegler et al.
Theochem. 2009
1
R
0  i0i0  ii
Ground state

3

0
i
Combining all terms
HF-case
Triplet excited state
0
a

HF
1. DEcyc = E(3 1 ) E(10 )  a0,HF  i0,HF  Kaa,i
i

TD-HF
1  i0a0  ia

HF
   a0,HF  i0,HF  Kaa,i
i
2.
Same result

KS-Case

TD-DFT
1. DEcyc = E(  )  E(  )  
3
2.
1
1
0
0,KS
a
  a0  i0  KaiKS,ai  KaiKS,i a

0,KS
i
1 KS 1 KS
KS
 Kaaaa  Ki i i i  Kaa,i
i
2
2
Different result
28
Coupling of a Single Occupied/Virtual Orbital Pair to all Orders.
Ziegler, et al. J. Chem.
Phys. 2008, 129, 184114
We consider the variation
0  12 ...i j ....n
"  12 ...i" j ....n
i"  cosi  sin a
The corresponding variation
in energy is
1
E "[ ]  E (0)  sin  sin  * [ a(0)   i(0)  K aiT ,ai ] [sin  sin   sin  * sin  * ]K aiT ,i a
2

1
1 T
[sin * sin  ]2 [ K iTi ,i i  K aa,aa
 K iTi ,aa  K aiT ,ai  K aiT ,i a ]
2
2
T=HF,KS
sin  sin  * [cos sin  *  cos * sin  ][K iTa,aa  K aiT ,i i ]
29
KS-Case


0
i
0
a
DPia 1 or ai  90
o
We have E "[ ]  E (0)  sin  sin  *[ a(0)   i(0)  K aiKS,ai ] 1 [sin  sin   sin  * sin  * ]K aiKS,i a

1 
1 KS
KS
KS
KS
[sin sin  ] [ K i i ,i i  K aa,aa  K iKS

K

K
]
i ,aa
ai ,ai
ai ,i a
2
2
sin  sin  * [cos sin  *  cos * sin  ][K iKS
 K aiKS,i i ]
a,aa
2
*
2
KS
KS
KS
KS
KS
Since Ki i ,i i  Kaa,aa  Kai ,i a  0 K ai ,ai  K aa,i i

KS
KS
KiKS

K

K
0
a,aa
ai ,i i
ai ,i a
We can not only retain second order terms as it is done in TD-DFT

(0) 
2 (0)
(0)  KS
E [ ] E  sin [    K ]
KS
For  ai  90o
KS
a
i
aa,i i
1
1 KS
"
(0)
KS
EKS
[ ] EKS
 [ KiKS

K

K
]
aa,aa
i i ,aa
2 i ,i i 2
31

A Revised Electronic Hessian for Approximate CV-DFT
We shall now attempt to construct a revised electronic ground state Hessian (A  B) R DFT
that can be used in eigenfunction equations where the required trust region is
UaiUai*  1, which includes Uai  1.

Ziegler, et al. J. Chem.
Phys. 2008, 129, 184114
We now consider

    ...  ....
"
"
1
"
2
"
i
"
j
"
n
With
vir
vir
occ
1
"
i  i  sin(ai )a  [cos(ai ) 1]i   [sin(ai )sin* (bk )k
2 ki b
a
a
It is readily shown that i"© is normalized to all orders in cosn ( ai )and sinm (ai ). It is also
orthonormal to first order in sin( ai )sin(bj ).

Keeping terms
in the energy to

4
4
 (ai ) ; sin (ai ) ; cos(ai )sin(bj )
cos
32
A Revised Electronic Hessian for CV-DFT
We now obtain the revised Hessian
"
0
E KS
 E KS
1 *
 R
2
A RDFT
R  RDFT
B
BRDFT R 
 * 
RDFT 
A
R 
where
KS
A[ai ]RDFT

d
d
[(



)]
K
ai,bj
ab ij
a
i
ai,bj
1
 sin(ai )* sin(ai )[K aa,aa  K ii,ii  2K ai,ia  2K ai,ai  2K aa,ii ]dabdij
2
[sin(ai )* cos(ai )  cos(ai )* sin(ai )][K aa,ai  K ai,ii ]dabdij
RDFT
KS
Bai,bj
 K ai,jb
Ziegler, et al. J. Chem.
Phys. 2008, 129, 184114
33
Revised KS-Case
A Revised Electronic Hessian for CV-DFT
Ria 1 or ai  90o
1
0  i0i0  ii
3
1  i0a0  ia

Ground state

0
i
0

 a
Triplet excited state
DE-KS
1.
1 KS 1 KS
3 1
1
0
0
0
KS
E(

)

E(

)





K

K

K
a i 2 aaaa 2 i i i i aa,i i

1 KS 1 KS
0
0
KS






K

K

K
a
aaaa
Revised CV-KS
Same result
i
aa,i i
2
2 iiii

34
Numerical example
R
Orbital optimization
1
1
DE  2 p  1s  K1sKS1s ,1s 1s  K2KSp2 p,2 p2 p  K1sKS1s ,2 p2 p
2
2
He
Li
Exp
I DSCF  ADSCF 1/ R
CV-DFT
I+A-1/R
TD-DFT
TD  DFT :   20 p  1s0  K2KSp1s ,2 p1s  K2KSp1s ,1s 2 p


35
Numerical example
Et TFEt
I DSCF  ADSCF 1/ R


CV-DFT
TD-DFT
36
Numerical example
I  A 1/ R
TFEtEt


CV-DFT
TD-DFT
37
Excited States
92nd Canadian Chemistry Conference
and
DFT-methods
Exhibition in Hamilton, ON, May 30 - June 3, 2009
1: DSCF (Slater, 1972)
Excited States
2: TD-DFT (Gross,Cassida 1995)
16:00 Tuesday June 2 2009
Heritage-Sheridan
Ground state
Excited States
92nd Canadian Chemistry Conference
and
DFT-methods
Exhibition in Hamilton, ON, May 30 - June 3, 2009
1: DSCF (Slater, 1972)
Excited States
2: TD-DFT (Gross,Cassida 1995)
Degeneracies
Slater Sum rules
Ziegler 1976
Spin-Restricted Open-Shell-KS
ROKS: Filatov 2000
Ground state
Excited States
92nd Canadian Chemistry Conference
and
DFT-methods
Exhibition in Hamilton, ON, May 30 - June 3, 2009
1: DSCF (Slater, 1972)
Excited States
2: TD-DFT (Gross,Cassida 1995)
Near Degeneracies
Ground state
Excited States
92nd Canadian Chemistry Conference
and
DFT-methods
Exhibition in Hamilton, ON, May 30 - June 3, 2009
1: DSCF (Slater, 1972)
Excited States
Near Degeneracies
I.Ensamble theory
Ziegler 1979
Filatov REKS (2002)
Ground state
II. Broken Symmetry
Fukotome (1973)
III. Density matrix
functional theory
Baerends (2009)
H2 Dissociation
 u u
 g u 
DuuS2   uu
T1   g u
 gg

DggS0   gg

Exact

1
 
{  g g   u u }
2
S0
Missing orbital optimization

0
1
2
3

4
H-H Bond Length (Å)
Dr. Mike Seth
Prof. Fan Wang
Dr. Mykhylo Krykunov
Prof. Jochen Autschbach
Concluding
Remarks
We have demonstrated that the electronic ground state Hessian (G DFT ) for
approximate DFT has a much smaller trust radius than the corresponding Hessian (G HF )

in HF-theory due t o an incomplete cancellation of Coulomb and exchange
self-interaction
DFT
terms in approximate T D-DFT. It is further point ed out that the use of G
in the basic
eigenvalue equations of T D-DFT is inconsistent with the trust radius of G DFT . We have

finally suggested a revised Hessian (G R  DFT ) for which the trust radius is similar t o that of

G HF and consistent with its use in T D-DFT . The new matrix G R DFT is expected to
improve the accuracy of 
excitation energies involving electron transitions between two

separated regions of space or between orbitals of different spatial extend.
46
Basic Time Dependent Density Functionl Theory
Basic Equation :
F (  )  W2 F (  )
M.E.Casida
TD-approximation AF(  )  W  F (  )
Where :
Gross,E.K.; Kohn W.

  S1/2 (A  B)S 1/2
S1/ 2  (A  B)1/ 2
Definition of A and B Matrices :

Aia, jb
  2E 
 
P P 

 ia jb 0
Bia,bj
  2E 
 
P P 

 ai bj 0
47
The Ground State Hessian
Energy change due to change in density away from ground state
E T [  0  D]  E T [  0 ]  DE T [D]  O(3)[E T (U)]
A T
1 *
 U U  T
2
B
BT U  (3)
 *  O [U]
T 
A U 
Change in density away from ground state
occ vir
D    Piai0a0
i
a
Ground state density
occ
 0  i0i0
i
48
The Ground State Hessian for HF
HF
E HF [  0  D]  E HF [ 0 ]  DE HF [D]  O(3)[E HF (U)]
A HF
1 *
 U U  HF
2
B
B HF U  (3)
 *  O [U]
HF 
A U 

Change in density away from ground state
K rsHF,tq  K rsC ,tq  K rsHF,tq,XC
1
 d r s d t q   (1)s (1) t (2)q* (2)d 1d 2
r12
1
*
d r t d s q  r (1)t (1) s (2)q* (2)d 1d 2
r12
*
r
49
The Ground State Hessian for HF
KS
E KS [  0  D ]  E KS [ 0 ]  DE HF [D]  O(3)[E KS (U)]
A KS
1 *
 U U  KS
2
B
B KS U  (3)
 *  O [U]
KS 
A U 
KS
C
Aai,bj
 dabdij (a  i )  Kai,bj
KS
ai,bj
Change in density away from ground stateB

K
KS
ai,bj
KS
C
KS,XC
K rs,tq
 K rs,tq
 K rs,tq
 d r s d t q   r* (1) s (1)
1
 t (2) q* (2)d1d 2
r12

d r t d s q   r* (1) t (1) f XC (1) s (1) q* (1)d1
50
Basic Ground State KS-theory
Normal Restrictions
Spin Symmetry Restrictions:
1
ˆssi (1)   i (1)
2
Space Symmetry Restrictions:

Ri   j D ji (R)
j
All KS-orbitals belongs
to a symmetry representation
characteristic for the point
group of the molecule

P
r
Not oall systems can be described by a single determinant:
b
l
Solutions subject to space and spin symmetry breaking:
e
m
Degenerate systems
Near degenerate systems
Similar Constraint in Wave Function Mechanics
In wave function mechanics we can write.
 123 .....ij .....n
where
vir
1 vir occ
i(1)  i (1) U aia (1)  U aiU ak* k (1) O(3) [U]
2 a k
a
by expanding in terms of determinants to second order
 (1
O (3) [U ]
1
U ai* U ai ) 0    U ai ai      U aiU bj i, ja,b


i
a
i
j
a
b
2 i a
Similar Constraint in Wave Function Mechanics
To second order
 (1
1
*
0
(3)
U
U
)

U


U
U

O
[U]








ai ai
ai ai
ai bj i,ja,b
i
a
i
a
i
j
a
b
2
The corresponding energy to second order is
E  (1   U ai* U ai )E 0   
i
 
i
j
a
U
a
b
  U U  Hˆ 
Hˆ       U U
i
U bj i, ja,b
ai
j
a
0
*
bj
ai
b
i
ai
j
a
b
b j
*
ai
*
bj

0 Hˆ i, ja,b O (3) [U]

For
(1   U ai* U ai )  0
i
 U U
a
i
a
*
ai
ai
1
E’ has no diagonal contribution from the ground state. It can be considered as the
energy of an excited state
