Mysovsky psi k2010-poster

Mysovsky psi k2010-poster
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  1. 1. I. Trivial example If w(r)=1 in entire space then g r functions are plane waves and eigenvalues are = 4 k 2 and -decomposition turns into well known Fourier representation of Coulomb interaction: 1 rr' = d k 2 3 4 k 2 exp[i krr'] II. Less trivial example With the weight function wr=[r r1 2 ] 1 - decomposition can be written analytically: (formula a) here is the angle between r and r' vectors, Pl are Legendre polynomials and Pn k are Jacobi polynomials. III. Another less trivial example Similarly, weight functions wr=[r 2 1 2 ] 1 leads to the formula (formula b) here is Euler's gamma function. Interesting feature of this formula the 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 -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 -decomposition: =d r d r' rrr'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 Ci g Ci g = = i Ki Ki , Ki = Ci g (Exchange part) V. How do g r functions look? Asymptotic behaviour of formula b functions: gn ,l=Cn ,l r l r 2 1 l 1 2 Pn l 1 2 r 2 1 r 2 1~ { r l r 0 1 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 -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 rr' 1 and that approximated by partial sum VII. Convergence of the integrals VIII. Relation to RI and Cholesky approaches Resolution of identity (RI): 1= n n n is used for decomposition of ERIs: = n ,m nnmm 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. Conclusion 1. A new mathematical result has been obtained. It is possible to construct spectral decomposition of rr' 1 . Moreover, there is infinite number of such decompositions 2. Two special cases of -decomposition have been constructed (formulas a and b). 3. The convergence of -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. -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. -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 operator A. Mysovsky e-mail: mysovsky@gmail.com Institute 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 wr f r where the eigenvalues form a discreet spectrum while the g r functions are normalized by the condition gg' w=d r wr g r g' r=' then the following decomposition takes place: 1 rr' = g r gr ' (-decomposition) n=10 n=20 n=30 n=50 exact 1 r n=10 1 rr' = n ,l=0 2l14l2n3 2ln12ln2 n!4ln2! [2ln1!]2 r l r12l1 Pn 2l1 r1 r1 r ' l r '12l1 Pn 2l1 r '1 r '1Pl cos 1 rr' = n ,l=0 4n!2ln1!2l1 [ln 3 2 ] 2 ln1 4ln12 1 r l r2 1 l 1 2 Pn l 1 2 r 2 1 r 2 1 r ' l r '2 1 l 1 2 Pn l 1 2 r ' 2 1 r ' 2 1Pl cos Ncutoff =20 Ngrid =50 Ncutoff =20 Ngrid =50 Ncutoff =50 Ngrid =100 Ncutoff =50 Ngrid =100