mysovsky psi k2010-poster
TRANSCRIPT
I. Trivial example
If w(r)=1 in entire space then g r functions are plane waves and eigenvalues are
=4
k 2
and λ-decomposition turns into well known Fourier representation of Coulomb interaction:
1∣r−r '∣
=∫ d k
23
4
k 2 exp [i k r−r ']
II. Less trivial example
With the weight function w r=[r r12 ]−1
λ-
decomposition can be written analytically:
(formula a)
here θ is the angle between r and r' vectors, Pl are
Legendre polynomials and Pnk are Jacobi
polynomials.
III. Another less trivial example
Similarly, weight functions w r=[r212 ]
−1 leads
to the formula
(formula b)
here Γ is Euler's gamma function. Interesting feature of this formula – the λ eigenvalue depends only on
ln1 , i.e. demonstrates the same type of
degeneracy as in quantum mechanical hydrogen atom problem.
IV. Hartree-Fock method with λ -decomposition
The electron repulsion integrals (ERIs) constituting the most time- and memory consuming part of Hatree-Fock calculation can be decomposed with the use of λ-decomposition:
∣=∫ d r d r 'rr r'r'
∣r-r'∣=
=∑
⟨∣g⟩ ⟨ g∣⟩
This leads to the decomposition of Coulomb and exchange parts of Fock operator:
J =2∑
P∣=
=2∑
⟨∣g⟩∑
P ⟨g∣⟩
(Coulomb part)
K =2∑
P∣=
∑ i
∑ C i ⟨∣g⟩∑ C i ⟨g∣⟩=
=∑ i
K i K i
, K i =∑
C i ⟨ g∣⟩
(Exchange part)
V. How do g r functions look?
Asymptotic behaviour of formula b functions:
gn , l=C n , lr l
r 21
l1
2
Pn
l1
2 r2
−1
r21 ~{
r l r01
r l1 r∞
(Several functions with l=0)with increasing radial number n the more and more nodes appear and these nodes tend to concentrate near the
point r=0.
VI. Convergence of λ -decomposition What happens if we cut the decomposition series on certain term?
(exact 1/r and formula b result with Ncutoff=10)
It's much more instructive to see the error, i.e. the
difference between exact ∣r−r '∣−1 and that
approximated by partial sum
VII. Convergence of the integrals
VIII. Relation to RI and Cholesky approachesResolution of identity (RI):
1=∑n
∣n ⟩ ⟨ n∣
is used for decomposition of ERIs:
∣=∑n , m
⟨∣n⟩n∣m⟨m∣⟩
Specially fitted auxiliary basis sets are required.Historically the first approach of avoiding calculation of ERIs in construction of the Fock matrix was based on Cholesky decomposition of integral array
∣=∑i
L
i L
i
The convergence of such decomposition was observed numerically but has never been proved.Conclusion1. A new mathematical result has been obtained. It is possible to construct spectral decomposition of
∣r−r '∣−1 . Moreover, there is infinite number of
such decompositions2. Two special cases of λ-decomposition have been constructed (formulas a and b).
3. The convergence of λ-decomposition has been investigated numerically. The decomposition is found to be rapidly converging except the point r=r'. In the vicinity of this point the convergence is nonuniform.
4. It is shown that relatively small number of terms of formula b allows to to reproduce ERIs with the error not exceeding 10-10.
5. λ-decomposition is closely related to other approaches of fast exchange calculation (RI and Cholesky decomposition based). In fact, it provides theoretical basis for understanding why these two approaches lead to convergent series.
6. λ-decomposition based algorithms can be applied not only to Hartree-Fock method, but to hybrid DFT and post-SCF correlated methods as well.
7. It allows the use of non-gaussian basis functions.
Spectral representation of Hartree-Fock exchange operatorA. Mysovskye-mail: [email protected] of geochemistry SB RAS, Irkutsk, Russia
Theorem
If w(r) is an arbitrary positively defined weight function and gr are bounded in entire space solutions of the equation
gr=−4
w r f r
where the eigenvalues λ form a discreet spectrum while the gr functions are normalized by the condition
⟨ g∣g ' ⟩w=∫ d r w r g
∗r g ' r= '
then the following decomposition takes place:
1∣r−r '∣
=∑
g
∗r gr ' (λ-decomposition)
n=10 n=20
n=30n=50
e x a c t 1r
n=10
1∣r−r '∣
=∑n , l=0
∞2l14l2n32ln12ln2
n! 4ln2!
[2ln1! ]2×
×r l
r12l1Pn
2l1 r−1r1 r ' l
r '12l1Pn
2l1 r '−1r '1 P l cos
1∣r−r '∣
=∑n , l=0
∞ 4n! 2ln1! 2l1
[ln3
2]
2
ln1
4 ln12−1×
×r l
r21l
12
Pn
l12 r 2
−1
r 21 r ' l
r ' 21l
12
Pn
l12 r ' 2
−1
r ' 21 P l cos
Ncutoff
=20
Ngrid
=50
Ncutoff
=20
Ngrid
=50 Ncutoff
=50
Ngrid
=100
Ncutoff
=50
Ngrid
=100