. Author manuscript; available in PMC: 2016 Feb 25.

Published in final edited form as: J Chem Theory Comput. 2007 Nov;3(6):2083–2097. doi: 10.1021/ct7001336

Table 1.

Indefinite integrals for pairwise descreening of multipoles through quadrupole

(l, m)₁

(l, m)₂

D_{(l, m)_i, (l, m)_i} (r_ij, R_j)

(0, 0)

- (2 ln (r) r^{2} + 4 r_{i j} r - r_{i j}^{2} + R_{j}^{2}) / 16 r_{i j} r^{2}

(1, 0)

- (4 r^{4} ln (r) + 4 r_{i j}^{2} r^{2} + 4 r^{2} R_{j}^{2} - r_{i j}^{4} + 2 r_{i j}^{2} R_{j}^{2} - R_{j}^{4}) / 64 r^{4} r_{i j}^{2}

(2, 0)

- (12 r^{6} ln (r) + 6 r_{i j}^{2} r^{4} + 18 r^{4} R_{j}^{2} + 3 r^{2} r_{i j}^{4} + 6 r^{2} r_{i j}^{2} R_{j}^{2} - 9 r^{2} R_{j}^{4} - 2 r_{i j}^{6} + 6 r_{i j}^{4} R_{j}^{2} - 6 r_{i j}^{2} R_{j}^{4} + 2 R_{j}^{6}) / 256 r_{i j}^{3} r^{6}

(1, 0)

- (12 r^{6} ln (r) - 42 r_{i j}^{2} r^{4} + 18 r^{4} R_{j}^{2} + 64 r^{3} r_{i j}^{3} - 21 r^{2} r_{i j}^{4} + 30 r^{2} r_{i j}^{2} R_{j}^{2} - 9 r^{2} R_{j}^{4} - 2 r_{i j}^{6} + 6 r_{i j}^{4} R_{j}^{2} - 6 r_{i j}^{2} R_{j}^{4} + 2 R_{j}^{6}) / 384 r^{6} r_{i j}^{3}

(2, 0)

- (24 r^{8} ln (r) - 48 r^{6} r_{i j}^{2} + 48 r^{6} R_{j}^{2} + 60 r_{i j}^{4} r^{4} + 72 r_{i j}^{2} r^{4} R_{j}^{2} - 36 r^{4} R_{j}^{4} - 16 r^{2} r_{i j}^{6} + 48 r^{2} r_{i j}^{4} R_{j}^{2} - 48 r^{2} r_{i j}^{2} R_{j}^{4} + 16 r^{2} R_{j}^{6} - 3 r_{i j}^{8} + 12 r_{i j}^{6} R_{j}^{2} - 18 r_{i j}^{4} R_{j}^{4} + 12 r_{i j}^{2} R_{j}^{6} - 3 R_{j}^{8}) / 1024 r^{8} r_{i j}^{4}

(1, 1)
(1, −1)

(12 r^{6} ln (r) + 102 r_{i j}^{2} r^{4} + 18 r^{4} R_{j}^{2} - 128 r^{3} r_{i j}^{3} + 51 r^{2} r_{i j}^{4} - 42 r^{2} r_{i j}^{2} R_{j}^{2} - 9 r^{2} R_{j}^{4} - 2 r_{i j}^{6} + 6 r_{i j}^{4} R_{j}^{2} - 6 r_{i j}^{2} R_{j}^{4} + 2 R_{j}^{6}) / 768 r^{6} r_{i j}^{3}

(2, 1)
(2, −1)

\sqrt{3} (24 r^{8} ln (r) + 96 r^{6} r_{i j}^{2} + 48 r^{6} R_{j}^{2} - 84 r_{i j}^{4} r^{4} - 72 r_{i j}^{2} r^{4} R_{j}^{2} - 36 r^{4} R_{j}^{4} + 32 r^{2} r_{i j}^{6} - 48 r^{2} r_{i j}^{4} R_{j}^{2} + 16 r^{2} R_{j}^{6} - 3 r_{i j}^{8} + 12 r_{i j}^{6} R_{j}^{2} - 18 r_{i j}^{4} R_{j}^{4} + 12 r_{i j}^{2} R_{j}^{6} - 3 R_{j}^{8}) / 3072 r^{8} r_{i j}^{4}

(2, 0)

- 3 (120 r^{10} ln (r) - 140 r^{8} r_{i j}^{2} + 300 r^{8} R_{j}^{2} - 540 r^{6} r_{i j}^{4} + 360 r^{6} r_{i j}^{2} R_{j}^{2} - 300 r^{6} R_{j}^{4} + 1024 r^{5} r_{i j}^{5} - 360 r^{4} r_{i j}^{6} + 600 r^{4} r_{i j}^{4} R_{j}^{2} - 440 r^{4} r_{j j}^{2} R_{j}^{4} + 200 r^{4} R_{j}^{6} - 35 r^{2} r_{i j}^{8} + 180 r^{2} r_{i j}^{6} R_{j}^{2} - 330 r^{2} r_{i j}^{4} R_{j}^{4} + 260 r^{2} r_{i j}^{2} R_{j}^{6} - 75 r^{2} R_{j}^{8} - 12 r_{i j}^{10} + 60 r_{i j}^{8} R_{j}^{2} - 120 r_{i j}^{6} R_{j}^{4} + 120 r_{i j}^{4} R_{j}^{6} - 60 r_{i j}^{2} R_{j}^{8} + 12 R_{j}^{10}) / 20480 r^{10} r_{i j}^{5}

(2, 1)
(2, −1)

(120 r^{10} ln (r) + 180 r^{8} r_{i j}^{2} + 300 r^{8} R_{j}^{2} + 900 r^{6} r_{i j}^{4} - 120 r^{6} r_{i j}^{2} R_{j}^{2} - 300 r^{6} R_{j}^{4} - 1536 r^{5} r_{i j}^{5} + 600 r^{4} r_{i j}^{6} - 680 r^{4} r_{i j}^{4} R_{j}^{2} - 120 r^{4} r_{i j}^{2} R_{j}^{4} + 200 r^{4} R_{j}^{6} + 45 r^{2} r_{i j}^{8} - 60 r^{2} r_{i j}^{6} R_{j}^{2} - 90 r^{2} r_{i j}^{4} R_{j}^{4} + 180 r^{2} r_{i j}^{2} R_{j}^{6} - 75 r^{2} R_{j}^{8} - 12 r_{i j}^{10} + 60 r_{i j}^{8} R_{j}^{2} - 120 r_{i j}^{6} R_{j}^{4} + 120 r_{i j}^{4} R_{j}^{6} - 60 r_{i j}^{2} R_{j}^{8} + 12 R_{j}^{10}) / 10240 r^{10} r_{i j}^{5}

(2, 2)
(2, −2)

(2, 2)
(2, −1)

- (120 r^{10} ln (r) + 1140 r^{8} r_{i j}^{2} + 300 r^{8} R_{j}^{2} - 4380 r^{6} r_{i j}^{4} - 1560 r^{6} r_{i j}^{2} R_{j}^{2} - 300 r^{6} R_{j}^{4} + 6144 r^{5} r_{i j}^{5} - 2920 r^{4} r_{i j}^{6} + 1880 r^{4} r_{i j}^{4} R_{j}^{2} + 840 r^{4} r_{i j}^{2} R_{j}^{4} + 200 r^{4} R_{j}^{6} + 285 r^{2} r_{i j}^{8} - 780 r^{2} r_{i j}^{6} R_{j}^{2} + 630 r^{2} r_{i j}^{4} R_{j}^{4} - 60 r^{2} r_{i j}^{2} R_{j}^{6} - 75 r^{2} R_{j}^{8} - 12 r_{i j}^{10} + 60 r_{i j}^{8} R_{j}^{2} - 120 r_{i j}^{6} R_{j}^{4} + 120 r_{i j}^{4} R_{j}^{6} - 60 r_{i j}^{2} R_{j}^{8} + 12 R_{j}^{10}) / 40960 r^{10} r_{i j}^{5}