Ab initio asymptotic-expansion coefficients for pair energies in Møller-Plesset perturbation theory for atoms

Transcription

1 Ab initio asymptotic-expansion coefficients for pair energies in Møller-Plesset perturbation theory for atoms K. JANKOWSKI a, R. SŁUPSKI a, and J. R. FLORES b a Nicholas Copernicus University Toruń, Poland b Universidad de Vigo Vigo, Spain

2 Outline OUTLINE 1 Slow-convergence problem of CI-type approaches. 2 Studies on the asymptotic expansion of second-order energies (He-like systems). 3 Second-order Møller-Plesset PT (MP2) for closed-shell atoms. 4 Asymptotic expansion (AE) of symmetry-adapted MP2 pair energies. 5 Disagreement of theoretical and computed AE coefficients. 6 Derivation of formulae for AE coefficients in the PW/m expansion. 7 Some examples. 8 Conclusions

3 Slow-convergence problem of CI-type approaches Slow convergence of many-electron wave functions in CI-type bases causes major problems in the description of correlation effects.

4 Slow-convergence problem of CI-type approaches Slow convergence of many-electron wave functions in CI-type bases causes major problems in the description of correlation effects. To overcome these problems reliable extrapolation methods are required in calculations of accurate energies. In the last decade much effort has been spent to develop methods of approximating the complete basis set (CBS) limit.

5 Slow-convergence problem of CI-type approaches Slow convergence of many-electron wave functions in CI-type bases causes major problems in the description of correlation effects. To overcome these problems reliable extrapolation methods are required in calculations of accurate energies. In the last decade much effort has been spent to develop methods of approximating the complete basis set (CBS) limit. The functions used in the energy extrapolation procedures are to some extend of empirical nature. However, there are several cases when guidance provided by formal mathematical results has been used.

6 Slow-convergence problem of CI-type approaches Slow convergence of many-electron wave functions in CI-type bases causes major problems in the description of correlation effects. To overcome these problems reliable extrapolation methods are required in calculations of accurate energies. In the last decade much effort has been spent to develop methods of approximating the complete basis set (CBS) limit. The functions used in the energy extrapolation procedures are to some extend of empirical nature. However, there are several cases when guidance provided by formal mathematical results has been used. Unfortunately, there is a real scarcity of such results. Until quite recently, formal information concerning the energy convergence was available only for He-like atomic systems.

7 Studies on the asymptotic expansion of second-order energies (He-like systems) C. Schwartz (1962), pioneered the studies on the rate of convergence correlation energies for atoms. For the He ground state within the 1/Z expansion Perturbation Theory (1/Z PT) the increments to the second-order energy E (2) l determined by orbitals corresponding to the angular momentum quantum number l decrease according to the asymptotic formula E (2) l = a 4 (l + 1/2) 4 + a 6 (l + 1/2) 6 + O [ (l + 1/2) 8],

8 Studies on the asymptotic expansion of second-order energies (He-like systems) C. Schwartz (1962), pioneered the studies on the rate of convergence correlation energies for atoms. For the He ground state within the 1/Z expansion Perturbation Theory (1/Z PT) the increments to the second-order energy E (2) l determined by orbitals corresponding to the angular momentum quantum number l decrease according to the asymptotic formula E (2) l = a 4 (l + 1/2) 4 + a 6 (l + 1/2) 6 + O [ (l + 1/2) 8], The basis-set truncation error, due to omission of functions of l > l max : δe (2) l max = l=l max +1 E (2) l l=l max +1/2 E (2) l dl = a 4 3 (l max + 1) 3 + a 6 5 (l max + 1) 5.

9 Studies on the asymptotic expansion of second-order energies (He-like systems) Kutzelnigg and Morgan (KM) 1992 generalized this result to arbitrary states of He-like systems in 1/Z PT.

10 Studies on the asymptotic expansion of second-order energies (He-like systems) Kutzelnigg and Morgan (KM) 1992 generalized this result to arbitrary states of He-like systems in 1/Z PT. They classified the singlet or triplet states two-electron systems with angular momentum L as natural- or unnatural-parity states: natural parity equal to ( 1) L, e.g., 1s2s 3 S, 2p3p 1 D, unnatural parity equal to ( 1) L+1, e.g., (2p) 2 3 P, 3p3d 1 D,

11 Studies on the asymptotic expansion of second-order energies (He-like systems) Kutzelnigg and Morgan (KM) 1992 generalized this result to arbitrary states of He-like systems in 1/Z PT. They classified the singlet or triplet states two-electron systems with angular momentum L as natural- or unnatural-parity states: natural parity equal to ( 1) L, e.g., 1s2s 3 S, 2p3p 1 D, unnatural parity equal to ( 1) L+1, e.g., (2p) 2 3 P, 3p3d 1 D, There are four types of states of He-like systems: 1 natural-parity singlets, 2 natural-parity triplets, 3 unnatural-parity triplets, 4 unnatural-parity singlets.

12 Studies on the asymptotic expansion of second-order energies (He-like systems) KATO S CORRELATION CUSP CONDITION ( ˆΨ r 12 )r 12 =0 = Ψ(r 12 = 0), where ˆΨ is Ψ averaged over a small sphere about R 12 = 0.

13 Studies on the asymptotic expansion of second-order energies (He-like systems) KATO S CORRELATION CUSP CONDITION ( ˆΨ r 12 )r 12 =0 = Ψ(r 12 = 0), where ˆΨ is Ψ averaged over a small sphere about R 12 = 0. First-order wave functions of He-like systems in 1/Z PT at r 12 = 0 Ψ (1) = α r 12Ψ (0) + O[r 2 12 ] = α (r 12) l P l (cos ϑ 12)Ψ (0) (1, 2) + O[r 2 12 ] (1) l=0 (r 12) l = 1 2l + 3 r l+2 < 1 r l+1 2l 1 > r l <, ϑ 12 angle between r 1 and r 2. r l 1 >

14 Studies on the asymptotic expansion of second-order energies (He-like systems) KATO S CORRELATION CUSP CONDITION ( ˆΨ r 12 )r 12 =0 = Ψ(r 12 = 0), where ˆΨ is Ψ averaged over a small sphere about R 12 = 0. First-order wave functions of He-like systems in 1/Z PT at r 12 = 0 Ψ (1) = α r 12Ψ (0) + O[r 2 12 ] = α (r 12) l P l (cos ϑ 12)Ψ (0) (1, 2) + O[r 2 12 ] (1) l=0 (r 12) l = 1 2l + 3 r l+2 < 1 r l+1 2l 1 > α = 1/2 natural-parity singlets (Kato,1957), r l <, ϑ 12 angle between r 1 and r 2. r l 1 > α = 1/4 natural- and unnatural-parity triplets (Pack, Byers-Brown, 1966), α = 1/6 unnatural-parity singlets (Kutzelnigg-Morgan, 1992).

15 Studies on the asymptotic expansion of second-order energies (He-like systems ASYMPTOTIC EXPANSIONS OF THE SECOND-ORDER ENERGY for He-like systems in 1/Z PT (K-M) 1 natural-parity singlets, E (2) l 2 natural-parity triplets, = a NS 4 (l + 1/2) 4 + a NS 6 (l + 1/2) 6 + O [ (l + 1/2) 8], E (2) l 3 unnatural-parity triplets, E (2) l 4 unnatural-parity singlets. E (2) l = a NT 6 (l + 1/2) 6 + O [ (l + 1/2) 8], = a UT 6 (l + 1/2) 6 + O [ (l + 1/2) 8], = a US 8 (l + 1/2) 8 + O [ (l + 1/2) 10], K-M derived formulae for a NS 4, ans 6, ant 6, a UT 6, a US 8 in terms of radial integrals. No comparison with results for computed energies made so far!

16 Studies on the asymptotic expansion of second-order energies (He-like systems Leading AE coefficients of the pair energies for He-like systems (KM notation) 1 a NS 4 = 3 8 NR(5) + F 0. [ 2 a NT 6 = N 3 a UT 6 = NF 1R (5) + [ 4 a US 8 = N N = (1 + δ ni n j δ li l j ) 1 F p [l 1, l 2] 2 min(l 1,l 2 ) n= F 0R (7) (F 1 G 1 ) R (5) F 1R (7) (F 2 G 2 ) R (5) + [ [n] n(n + 1) Radial integrals: R (5) + = ] p { l1 l 1 n l 2 l 2 L 0 ] ] }( l1 l 1 n ρ +(r, r)r 5 dr, R (7) = )( l2 l 2 n ρ (r, r)r 7 dr, where ρ ±(r, r ) = 1 Rn1 l 2 1 (r)r n2 l 2 (r ) ± R n1 l 1 (r )R n2 l 2 (r) 2 ρ (r, r ) = 2 ρ (r, r ), r 2 R ni l i (r) radial parts of hydrogenlike orbitals. )

17 Second-order Møller-Plesset PT (MP2) for closed-shell atoms Second-order Møller-Plesset perturbation theory (MP2) in Symmetry Adapted Pair Functions (SAPF) formulation H = N h(i) + r 1 ij, H 0 H HF = N f (i) i=1 i<j i=1 h(i) bare nuclear one-electron Hamiltonian N Fock operator: f (i) = h(i) + [J n(i) K n(i)], n=1 J n(i) and K n(i) Coulomb and exchange potentials

18 Second-order Møller-Plesset PT (MP2) for closed-shell atoms Second-order Møller-Plesset perturbation theory (MP2) in Symmetry Adapted Pair Functions (SAPF) formulation H = N h(i) + r 1 ij, H 0 H HF = N f (i) i=1 i<j i=1 h(i) bare nuclear one-electron Hamiltonian N Fock operator: f (i) = h(i) + [J n(i) K n(i)], n=1 J n(i) and K n(i) Coulomb and exchange potentials Perturbation operator: V = H H HF = i<j r 1 ij N n=1 N i=1 [J n(i) K n(i)], Ψ (0) = Φ HF consists of canonical HF spinorbitals f (1)ϕ k (1) = ɛ k ϕ k (1), (k = 1,..., N).

19 Second-order Møller-Plesset PT (MP2) for closed-shell atoms Equation for Ψ (1) (N-electron level) : (H HF E 0 )Ψ (1) + (V E (1) )Ψ HF = 0, Brillouin theorem Ψ (1) consists entirely of double excitations on Φ HF. Ψ (1) (1, 2,..., N) = 2 1/2 Â[u ij (i, j) ϕ k (k)], i<j k i,j  and ˆB N- and 2-electron antisymmetrizers, respectively. First-order spinorbital pair functions u ij - - strongly orthogonal to spinorbitals of Ψ HF : Ω(1, 2)u ij (1, 2) = u ij (1, 2), Ω(1, 2) strong-orthogonality projector Ω(1, 2)[J n(i) K n(i)]ψ HF = 0 Ω(1, 2)V Ψ HF = Ω(1, 2)r 1 12 Ψ HF

20 Second-order Møller-Plesset PT (MP2) for closed-shell atoms Equation for Ψ (1) (N-electron level) : (H HF E 0 )Ψ (1) + (V E (1) )Ψ HF = 0, Brillouin theorem Ψ (1) consists entirely of double excitations on Φ HF. Ψ (1) (1, 2,..., N) = 2 1/2 Â[u ij (i, j) ϕ k (k)], i<j k i,j  and ˆB N- and 2-electron antisymmetrizers, respectively. First-order spinorbital pair functions u ij - - strongly orthogonal to spinorbitals of Ψ HF : Ω(1, 2)u ij (1, 2) = u ij (1, 2), Ω(1, 2) strong-orthogonality projector Ω(1, 2)[J n(i) K n(i)]ψ HF = 0 Ω(1, 2)V Ψ HF = Ω(1, 2)r 1 12 Ψ HF Equations for Ψ (1) (2-electron level) : [f (1) + f (2) ɛ i ɛ j ]u ij (1, 2) + Ω(1, 2)r 1 12 ˆB[ϕ i (1)ϕ j (2)] = 0,

21 Second-order Møller-Plesset PT (MP2) for closed-shell atoms First-order symmetry-adapted pair functions (SAPF): u(t 1, 2) = ˆB [R T (r 1, r 2)Z l i l j (LM L δ 1, δ 2) S(SM S 1, 2)], where T (T, M L, M S ) with T (n i l i, n j l j, L, S), ( ) Z j 1 j 2 j (LM L δ 1, δ 2) 1 j 2 +M [L] 1/2 j1 j 2 L Y m 1 (δ m 1 m 2 M j 1 1)Y m 2 (δ j 2 2). m 1,m 2 ( 1)

22 Second-order Møller-Plesset PT (MP2) for closed-shell atoms First-order symmetry-adapted pair functions (SAPF): u(t 1, 2) = ˆB [R T (r 1, r 2)Z l i l j (LM L δ 1, δ 2) S(SM S 1, 2)], where T (T, M L, M S ) with T (n i l i, n j l j, L, S), ( ) Z j 1 j 2 j (LM L δ 1, δ 2) 1 j 2 +M [L] 1/2 j1 j 2 L Y m 1 (δ m 1 m 2 M j 1 1)Y m 2 (δ j 2 2). Equations for u(t 1, 2): m 1,m 2 ( 1) [f (1) + f (2) ɛ ni l i ɛ nj l j ]u(t 1, 2) + Ω(1, 2)r 1 12 u0(t 1, 2) = 0. u 0(T 1, 2) = ˆB [R ni l i (r 1)R nj l j (r 2)Z l i l j (LM L δ 1, δ 2) S(SM S 1, 2)] R nj l j (r) HF orbitals.

23 Second-order Møller-Plesset PT (MP2) for closed-shell atoms First-order symmetry-adapted pair functions (SAPF): u(t 1, 2) = ˆB [R T (r 1, r 2)Z l i l j (LM L δ 1, δ 2) S(SM S 1, 2)], where T (T, M L, M S ) with T (n i l i, n j l j, L, S), ( ) Z j 1 j 2 j (LM L δ 1, δ 2) 1 j 2 +M [L] 1/2 j1 j 2 L Y m 1 (δ m 1 m 2 M j 1 1)Y m 2 (δ j 2 2). Equations for u(t 1, 2): m 1,m 2 ( 1) [f (1) + f (2) ɛ ni l i ɛ nj l j ]u(t 1, 2) + Ω(1, 2)r 1 12 u0(t 1, 2) = 0. u 0(T 1, 2) = ˆB [R ni l i (r 1)R nj l j (r 2)Z l i l j (LM L δ 1, δ 2) S(SM S 1, 2)] R nj l j (r) HF orbitals. Main partial wave (PW/m) expansion of u(t 1, 2) u(t 1, 2) = u j1j2 (T 1, 2) : j 1 j 2 u j1j2(t 1, 2) = ˆB [R T j 1 j 2 (r 1, r 2)Z j 1 j 2 (LM L δ 1, δ 2) S(SM S 1, 2)].

24 Second-order Møller-Plesset PT (MP2) for closed-shell atoms MP2 energy of an N-electron atom (SAPF formulation) E (2) = T E (2) (T ); Second-order pair energy for the SAP specified by T : E (2) (T ) = u(t ) Ωr 1 12 u 0(T ). PW/m expansion of the second-order SAP energy: E (2) (T ) = j 1j 2 E (2) j 1j 2 (T ), E (2) j 1j 2 (T ) Second-order PW/m energy increment: E (2) j 1j 2 (T ) = u j1j2 (T ) Ωr 1 12 u 0(T )

25 Second-order Møller-Plesset PT (MP2) for closed-shell atoms Equations for the first-order SAPFs He-like systems 1/Z PT [h(1) + h(2) E ni l i E nj l j ]Ψ (1) (T 1, 2) + [r 1 12 E (1) ]Ψ (0) (T 1, 2) = 0, Electron pairs in atoms MP2 PT [f (1) + f (2) ɛ ni l i ɛ nj l j ]u(t 1, 2) + Ω(1, 2)r 1 12 u0(t 1, 2) = 0. Zero-order SAPFs: Ψ 0 (T 1, 2) = ˆB [ R H n i l i (r 1) R H n j l j (r 2)Z l i l j (LM L δ 1, δ 2) S(SM S 1, 2)]. u 0(T 1, 2) = ˆB [R ni l i (r 1)R nj l j (r 2)Z l i l j (LM L δ 1, δ 2) S(SM S 1, 2)]. R H n j l j (r) and R nj l j (r) hydrogenlike and HF radial orbitals, respectively.

26 Second-order Møller-Plesset PT (MP2) for closed-shell atoms Relevant differences between 1/Z-PT and MP-PT equations: 1 h(i) f (i) (no singular terms introduced by J n and K n terms) 2 r 1 12 Ω(1, 2)r 1 12 (Ω(1, 2) modifies, for any atom, only terms with l 3 in the PW expansion of r 1 12 ).

27 Second-order Møller-Plesset PT (MP2) for closed-shell atoms Relevant differences between 1/Z-PT and MP-PT equations: 1 h(i) f (i) (no singular terms introduced by J n and K n terms) 2 r 1 12 Ω(1, 2)r 1 12 (Ω(1, 2) modifies, for any atom, only terms with l 3 in the PW expansion of r 1 12 ). In the vicinity of r 12 = 0 (i.e., for l 3) Ω(1, 2)r 1 12 r 1 12, In vicinity of r 12 = 0, u(t 1, 2) should have an analogous form like Ψ (1), i.e., u(t 1, 2) α r 12 u 0 (T 1, 2) + O[r12 2 ] = ũ l (T 1, 2) + O[r12 2 ] l=0 where ũ l (T 1, 2) α (r 12 ) l P l (cos ϑ 12 ) u 0 (T 1, 2). (2) K-M formulae for the AE coefficients should be applicable after replacing the hydrogenic radial orbitals by the HF ones. Possibility of comparison of theoretical and computed values of AE coefficients

28 Disagreement of theoretical and computed AE coefficients Flores-Słupski-Jankowski-Malinowski (JCP 121,12334(2004)) For Zn symmetry adapted pairs (SAP) PW/m up to l=45! 28 natural-parity singlets 18 natural-parity triplets 7 unnatural-parity triplets 3 unnatural-parity singlets Very accurate AE coefficients a 1 and a 2 have been computed by a specially designed method from accurate E (2) j 1 j 2 (T ) PW/m increments (JCP 124,104107(2006)).

29 Disagreement of theoretical and computed AE coefficients Flores-Słupski-Jankowski-Malinowski (JCP 121,12334(2004)) For Zn symmetry adapted pairs (SAP) PW/m up to l=45! 28 natural-parity singlets 18 natural-parity triplets 7 unnatural-parity triplets 3 unnatural-parity singlets Very accurate AE coefficients a 1 and a 2 have been computed by a specially designed method from accurate E (2) j 1 j 2 (T ) PW/m increments (JCP 124,104107(2006)). We obtained AE coefficients which agree with the formulae: 1 a NS 4 = 3 8 NR(5) + F0. 2 a NT 6 = N [ 5 64 F0R(7) ( a UT 6 = ( 7 3 ) NF1R(5) + [ 4 a US 8 = N ( 9 5 ) F1R(7) + 77 ] ) (F1 G1) R(5) ( 9 2 ] ) (F2 G2) R(5) +

30 Disagreement of theoretical and computed AE coefficients Flores-Słupski-Jankowski-Malinowski (JCP 121,12334(2004)) For Zn symmetry adapted pairs (SAP) PW/m up to l=45! 28 natural-parity singlets 18 natural-parity triplets 7 unnatural-parity triplets 3 unnatural-parity singlets Very accurate AE coefficients a 1 and a 2 have been computed by a specially designed method from accurate E (2) j 1 j 2 (T ) PW/m increments (JCP 124,104107(2006)). We obtained AE coefficients which agree with the formulae: 1 a NS 4 = 3 8 NR(5) + F0. 2 a NT 6 = N [ 5 64 F0R(7) ( a UT 6 = ( 7 3 ) NF1R(5) + [ 4 a US 8 = N ( 9 5 ) F1R(7) + 77 ] ) (F1 G1) R(5) ( 9 2 ] ) (F2 G2) R(5) + WHAT IS THE REASON OF THIS DISAGREEMENT?

31 Derivation of formulae for AE coefficients in the PW/m expansion The MP2 SAP energies in the large-l regime E (2) (T ) = u(t ) Ωr 1 12 u 0(T ) = αr 12 u 0 (T ) }{{} α l=0 (r 12) l P l (cos ϑ 12 ) u 0 (T ) Ωr 1 12 u 0(T ) + O [ (l + 1/2) 8]

32 Derivation of formulae for AE coefficients in the PW/m expansion The MP2 SAP energies in the large-l regime E (2) (T ) = u(t ) Ωr 1 12 u 0(T ) = αr 12 u 0 (T ) }{{} α l=0 (r 12) l P l (cos ϑ 12 ) u 0 (T ) E (2) can be represented by the PW/a expansion: E (2) (T ) = E (2) (T ), l l=0 Ωr 1 12 u 0(T ) + O [ (l + 1/2) 8] [ E (2) l (T ) = α (r 12 ) l P l (cos ϑ 12 ) u 0 (T 1, 2) Ωr }{{} 1 12 u 0(T ) + O (l + 1/2) 8], ũ l (T 1,2) For l 3 we have: E (2) (T ) = ũ l l (T ) r 1 12 u 0(T ),

33 Derivation of formulae for AE coefficients in the PW/m expansion Correspondence of PW/m and PW/a increments E (2) j 1 j 2 (T ) and E (2) l (T ) Consider the angular parts of ũ l (T 1, 2) and u j1 j 2 (T 1, 2). A useful relation: P k (cos ϑ 12) }{{} = l 4π l l Y m l (δ 1 )Y m (δ l 2 ) c w v 1 v 2 (t 1, t 2, L) ( 1) w+l [t 1, t 2, v 1, v 2] 1/2 { v1 v 2 L t 2 t 1 w The energy increment in the PW/a expansion for l 3: Z t 1 t 2 k (LM L δ 1, δ 2) = cv 1 v (t 2 1, t 2, L) Z v 1 v 2 (LM L δ 1, δ 2). v 1 v 2 E (2) (T ) = α (r l 12) l P l (cos ϑ 12)u 0(T ) r 1 12 u0(t ) = }( t1 v 1 w )( t2 v 2 w l cj 1 j (T )E l j 2 1 j 2 (T ), ) j 1 j 2 E l j 1 j 2 (T ) v l j 1 j 2 (T ) r 1 12 u 0(T ) v l j 1 j 2 (T 1, 2) α ˆB [(r 12) l R ni l i (r 1)R nj l j (r 2)Z j 1 j 2 (LM L δ 1, δ 2) S(SM S 1, 2)].

34 Derivation of formulae for AE coefficients in the PW/m expansion The lth PW/a increment is a sum of constituents of several PW/m increments. For l min 3: E (2) j 1j 2 (T ) (j 1 j 2) l min E (2) l (T ), l=l min Rearranging the terms on rhs we obtain: E (2) j 1j 2 (T ) min(j 1+l i,j 2+l j ) l=max( j 1 l i, j 2 l j ) c l j 1j 2 (T )E l j 1j 2 (T ) (l i and l j defined by T ). (3)

35 Some examples EXAMPLE: Unnatural-parity triplet state (np) 2 3 P The only nonzero indices are: c l l 1 l 1 and cl l+1 l+1. PW/a energy increment: E (2) l = c l l 1 l 1E l l 1 l 1 + c l l+1 l+1e l l+1 l+1 The l-th term of PW/a consists of the (l 1)-th, and (l + 1)-th terms in PW/m. PW/m energy increment: E (2) ll c l 1 ll E l 1 ll + c l+1 ll E l+1 ll.

36 Some examples Extrapolation procedure To derive the formulae for the AE coefficients, extract from Ẽ (2) l (T ) and E l j 1 j 2 (T ), the coefficient of the lowest-k term in the (l + 1/2) k expansion. For the PW/a expansion we can to some extend follow the extrapolation procedure developed by KM. Our approach is different from that of Kutzelnigg and Morgan: We consider directly the pair energies E (2) (T ). KM studied the Hylleraas functional E[Φ] = Φ H (0) E (0) Φ + 2 Re Φ V E (1) Ψ (0), which mplies dealing with kinetic energy terms that mask the simple structure of the AE coefficients.

37 Some examples Comparison of ab initio (a HF 1 ) and numerical (a 1) AE coefficients for Ne (in units of E h ). Pair a HF 1 a 1 1s 2 1 S s2s 1 S s2s 3 S s 2 1 S s2p 1 P s2p 3 P s2p 1 P s2p 3 P p 2 1 S p 2 1 D p 2 3 P

38 Summary Summary Relationship between the second-order He-like 1/Z-PT and Møller-Plesset PT (MP2) theory has been discussed.

39 Summary Summary Relationship between the second-order He-like 1/Z-PT and Møller-Plesset PT (MP2) theory has been discussed. The relevance of using two partial-wave expansions: PW/a and PW/m (directly related to the CI expansion), in asymptotic-expansion (AE) studies is recognized.

40 Summary Summary Relationship between the second-order He-like 1/Z-PT and Møller-Plesset PT (MP2) theory has been discussed. The relevance of using two partial-wave expansions: PW/a and PW/m (directly related to the CI expansion), in asymptotic-expansion (AE) studies is recognized. Equations relating the PW/m and PW/a energy increments are derived.

41 Summary Summary Relationship between the second-order He-like 1/Z-PT and Møller-Plesset PT (MP2) theory has been discussed. The relevance of using two partial-wave expansions: PW/a and PW/m (directly related to the CI expansion), in asymptotic-expansion (AE) studies is recognized. Equations relating the PW/m and PW/a energy increments are derived. Similarity of the formulae for the PW/a AE coefficients of the MP2 pair energies with those for the He-like 1/Z-PT expansion is justified.

42 Summary Summary Relationship between the second-order He-like 1/Z-PT and Møller-Plesset PT (MP2) theory has been discussed. The relevance of using two partial-wave expansions: PW/a and PW/m (directly related to the CI expansion), in asymptotic-expansion (AE) studies is recognized. Equations relating the PW/m and PW/a energy increments are derived. Similarity of the formulae for the PW/a AE coefficients of the MP2 pair energies with those for the He-like 1/Z-PT expansion is justified. Formulae for AE coefficients of the PW/m expansion are derived.

43 Summary Summary Relationship between the second-order He-like 1/Z-PT and Møller-Plesset PT (MP2) theory has been discussed. The relevance of using two partial-wave expansions: PW/a and PW/m (directly related to the CI expansion), in asymptotic-expansion (AE) studies is recognized. Equations relating the PW/m and PW/a energy increments are derived. Similarity of the formulae for the PW/a AE coefficients of the MP2 pair energies with those for the He-like 1/Z-PT expansion is justified. Formulae for AE coefficients of the PW/m expansion are derived. Theoretical PW/m AE coefficients showed close agreement with those extracted from computed energy increments.

44 Summary Summary Relationship between the second-order He-like 1/Z-PT and Møller-Plesset PT (MP2) theory has been discussed. The relevance of using two partial-wave expansions: PW/a and PW/m (directly related to the CI expansion), in asymptotic-expansion (AE) studies is recognized. Equations relating the PW/m and PW/a energy increments are derived. Similarity of the formulae for the PW/a AE coefficients of the MP2 pair energies with those for the He-like 1/Z-PT expansion is justified. Formulae for AE coefficients of the PW/m expansion are derived. Theoretical PW/m AE coefficients showed close agreement with those extracted from computed energy increments. Example of applications of the PW/m coefficients is presented.

45 Summary Summary Relationship between the second-order He-like 1/Z-PT and Møller-Plesset PT (MP2) theory has been discussed. The relevance of using two partial-wave expansions: PW/a and PW/m (directly related to the CI expansion), in asymptotic-expansion (AE) studies is recognized. Equations relating the PW/m and PW/a energy increments are derived. Similarity of the formulae for the PW/a AE coefficients of the MP2 pair energies with those for the He-like 1/Z-PT expansion is justified. Formulae for AE coefficients of the PW/m expansion are derived. Theoretical PW/m AE coefficients showed close agreement with those extracted from computed energy increments. Example of applications of the PW/m coefficients is presented.

46 Summary Summary Relationship between the second-order He-like 1/Z-PT and Møller-Plesset PT (MP2) theory has been discussed. The relevance of using two partial-wave expansions: PW/a and PW/m (directly related to the CI expansion), in asymptotic-expansion (AE) studies is recognized. Equations relating the PW/m and PW/a energy increments are derived. Similarity of the formulae for the PW/a AE coefficients of the MP2 pair energies with those for the He-like 1/Z-PT expansion is justified. Formulae for AE coefficients of the PW/m expansion are derived. Theoretical PW/m AE coefficients showed close agreement with those extracted from computed energy increments. Example of applications of the PW/m coefficients is presented. The present results provide an relationship between the Hartree-Fock wave function and the asymptotic behavior of the MP2 correlation energy.