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: mysovsky@gmail.comInstitute 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