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PAMOC RV 2014-03-01 PAMOC
Stewart partitioning of the Electron Density
in fuzzy atomic components (pseudoatoms)
<LTLALA> 23 K; Hpol 27.6.92; Uij H (ADPH 8.6.93) <LTLALA>
M.Barzaghi@istm.cnr.it pamoc.cloudvent.net
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Conversion factors of multipole moments
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from atomic units from atomic units from A^l
to standard units to A^l to standard units
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Dipole 2.54177 Debye 0.52918 A 4.80324
Quadrupole 1.34504 Debye-Ang = Buckingham 0.28003 A^2 4.80324
Octupole 0.71177 Debye-Ang**2 0.14818 A^3 4.80324
Hexadecapole 0.37665 Debye-Ang**3 0.07842 A^4 4.80324
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Volume integration for atom C1
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TABLE I - Spherical harmonic multipole moments of order l = 0, 1, 2, 3, 4.
_________________________________________________________________________________________________________________________________________________________________________
| | | __ | __ | __ | | | | |
| | | / | / | / | | | | |
| | | / A(l,m)dV = | / A(l,m)**2 dV = | / |A(l,m)|dV = | | | | |
VALRAY| | |__/ |__/ | __/ | | | | |
Pop's| <A(l,m)*r**l>| C(l,m)|4*pi*d(l,0)*d(m,0)|[M(l,m)*C(l,m)]^(-2)| [L(l,m)*C(l,m)]^(-1)| 1/M(l,m)^2 | M(l,m) | 1/L(l,m) | L(l,m) |
________________|___________________________|_______|__________________|____________________|_____________________|______________|____________|____________|____________|
| | | | | | | | | |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) |
________________|___________________________|_______|__________________|____________________|_____________________|______________|____________|____________|____________|
Core 2.0243 1.0 12.566370611 12.566370611 12.566370611 12.566370611 0.28209479 12.56637061 0.07957747
Valence 4.1392 1.0 12.566370611 12.566370611 12.566370611 12.566370611 0.28209479 12.56637061 0.07957747
(0, 0) 6.1635 <q> 6.1635 1.0 12.566370611 12.566370611 12.566370611 12.566370611 0.28209479 12.56637061 0.07957747
l = 1 N(l) = alpha(l)**[n(l)+l+3]/[n(l)+l+2]! = 628.8768 alpha(l) = 6.5007 (1/A) n(l) = 2
(1,+1) 0.0349 <x> -0.0116 1.0 0.000000000 4.188790204 6.276259859 4.188790204 0.48860251 6.27625986 0.15933056
(1,-1) -0.0406 <y> 0.0135 1.0 -0.000000000 4.188790204 6.276259859 4.188790204 0.48860251 6.27625986 0.15933056
(1, 0) 0.0250 <z> -0.0083 1.0 -0.000000000 4.188790204 6.276259859 4.188790204 0.48860251 6.27625986 0.15933056
l = 2 N(l) = alpha(l)**[n(l)+l+3]/[n(l)+l+2]! = 681.3529 alpha(l) = 6.5007 (1/A) n(l) = 2
(2,+2) 0.3963 <x2-y2> -0.1057 3.0 0.000000000 3.351032163 5.282328658 30.159289465 0.18209141 15.84698597 0.06310348
(2,-2) -1.6884 <xy> 0.1126 6.0 0.000000000 0.837758041 2.657537782 30.159289469 0.18209141 15.94522669 0.06271469
(2,+1) -0.2541 <xz> 0.0169 3.0 0.000000000 0.837758041 2.657537782 7.539822367 0.36418281 7.97261335 0.12542939
(2,-1) 0.4418 <yz> -0.0295 3.0 0.000000000 0.837758041 2.657537782 7.539822367 0.36418281 7.97261335 0.12542939
(2, 0) 0.1823 <z2-r2/3> -0.0162 1.5 0.000000250 1.117010721 3.222584082 2.513274122 0.63078313 4.83387612 0.20687332
l = 3 N(l) = alpha(l)**[n(l)+l+3]/[n(l)+l+2]! = 514.1617 alpha(l) = 6.5007 (1/A) n(l) = 3
(3,+3) -0.1250 <x(x2-3y2)> 0.0286 15.0 0.000000000 2.872313283 4.691966698 646.270488575 0.03933624 70.37950047 0.01420868
(3,-3) -0.5285 <y(3x2-y2)> 0.1208 15.0 -0.000000000 2.872313283 4.691966698 646.270488575 0.03933624 70.37950047 0.01420868
(3,+2) 0.3212 <z(x2-y2)> -0.0122 15.0 0.000000000 0.478718880 1.969009371 107.711748104 0.09635371 29.53514057 0.03385797
(3,-2) -3.4011 <xyz> 0.0324 30.0 0.000000000 0.119679720 0.992264388 107.711748105 0.09635371 29.76793164 0.03359320
(3,+1) -1.1478 <x(5z2-r2)> 0.4373 1.5 -0.000000000 4.787188804 6.241163913 10.771174810 0.30469720 9.36174587 0.10681768
(3,-1) 0.8046 <y(5z2-r2)> -0.3065 1.5 -0.000000000 4.787188804 6.241163913 10.771174810 0.30469720 9.36174587 0.10681768
(3, 0) -0.5150 <z(5z2-3r2)> 0.2943 0.5 -0.000000000 7.180783206 8.137588892 1.795195802 0.74635267 4.06879445 0.24577304
l = 4 N(l) = alpha(l)**[n(l)+l+3]/[n(l)+l+2]! = 241.4198 alpha(l) = 6.5007 (1/A) n(l) = 4
(4,+4) -0.5627 <x4-6x2y2+y4> 0.1143 105.0 -0.000000000 2.553167362 4.244354409 28148.670161895 0.00596034 445.65721298 0.00224388
(4,-4) 1.2686 <xy(x2-y2)> -0.0161 420.0 0.000000000 0.159572960 1.044537908 28148.670175494 0.00596034 438.70592143 0.00227943
(4,+3) -0.9524 <xz(x2-3y2)> 0.0242 105.0 0.000000000 0.319145920 1.589594624 3518.583771468 0.01685839 166.90743557 0.00599134
(4,-3) -0.5768 <yz(3x2-y2)> 0.0146 105.0 0.000000000 0.319145920 1.589594624 3518.583771468 0.01685839 166.90743557 0.00599134
(4,+2) -0.0266 <(x2-y2)(7z2-r2)> 0.0095 7.5 0.000000000 4.468042883 5.930668747 251.327412183 0.06307831 44.48001560 0.02248201
(4,-2) -0.5528 <xy(6z2-x2-y2)> 0.0491 15.0 0.000000000 1.117010721 3.009768589 251.327412243 0.06307831 45.14652883 0.02215010
(4,+1) 0.2469 <xz(4z2-3x2-3y2)> -0.0439 2.5 0.000000000 2.234021442 4.207032384 13.962634015 0.26761862 10.51758096 0.09507890
(4,-1) -0.0440 <yz(4z2-3x2-3y2)> 0.0078 2.5 0.000000000 2.234021442 4.207032384 13.962634015 0.26761862 10.51758096 0.09507890
(4, 0) 0.2512 <7z4-6z2r2+3r4/5> -0.0715 0.6 0.000000299 3.574434306 5.744978698 1.396263401 0.84628438 3.59061169 0.27850408
_________________________________________________________________________________________________________________________________________________________________________
(1) VALRAY population coefficients. Units are A^L.
(2) Expectation values <A(l,m,p)*r**l>, calculated by numerical volume integration according to equation:
__
/
<A(l,m)*r**l> = / A(l,m)*(r**l)*Rho(V)*dV
__/
V
with l >= 0 and -l <= m <= l, and the integral is over the volume V of the charge distribution.
The A(l,m)'s are defined according to the VALRAY subroutine ang.f. Units are A^L.
Values of <A(l,m)*r**l> can be obtained from the population coefficients by equation: (2) = - (1) * (5) / 4*pi.
(3) Factors C(l,m) convert VALRAY angular functions A(l,m) into the corresponding un-normalized associate Legendre polynomials times cos(|m|p) or sin(|m|p).
(4) Integral (over the solid angle) of the angular functions used in the VALRAY code. It is equal to 4*pi * d(l,0) * d(m,0), where d(i,j) is the Kronecker delta function.
(5) Integral (over the solid angle) of the squared value of the angular functions used in the VALRAY code. It is equal to 1/[M(l,m) * C(l,m)]**2.
(6) Integral (over the solid angle) of the absolute value of the angular functions used in the VALRAY code. It is equal to [2 - d(l,0)]/[L(l,m) * C(l,m)].
(7)-(8) The M(l,m)'s are wavefunction-normalization factors of real spherical harmonics.
(9)-(10) The L(l,m)'s are density-normalization factors of real spherical harmonics.
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TABLE II - Unabridged and traceless cartesian electrostatic
multipole moments (atomic units)
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Unabridged Traceless Tracelesss Traceless
from direct from direct from from spherical
numerical numerical unabridged harmonic
integration integration moments moments
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<q> = 6.1635 6.1635 6.1635 6.1635
<x> = -0.0220 -0.0220 -0.0220 -0.0220
<y> = 0.0256 0.0256 0.0256 0.0256
<z> = -0.0158 -0.0158 -0.0158 -0.0158
<xx> = -4.9151 -0.2396 -0.2396 -0.2396
<xy> = 0.4020 0.6029 0.6029 0.6029
<yy> = -4.5377 0.3264 0.3264 0.3264
<xz> = 0.0605 0.0907 0.0907 0.0907
<yz> = -0.1052 -0.1578 -0.1578 -0.1578
<zz> = -4.8132 -0.0868 -0.0868 -0.0868
<xxx> = -0.4413 -0.9861 -0.9861 -0.9861
<yyy> = 0.1609 0.2661 0.2661 0.2661
<zzz> = 0.3636 0.9930 0.9930 0.9930
<xyy> = -0.2113 -0.4893 -0.4893 -0.4893
<xxy> = 0.3254 0.7680 0.7680 0.7680
<xxz> = -0.2511 -0.5997 -0.5997 -0.5997
<xzz> = 0.5745 1.4754 1.4754 1.4754
<yzz> = -0.3955 -1.0342 -1.0342 -1.0342
<yyz> = -0.1685 -0.3933 -0.3933 -0.3933
<xyz> = 0.2186 0.5465 0.5465 0.5465
<xxxx> = -21.3634 0.5460 0.5460 0.5460
<yyyy> = -19.8156 0.6214 0.6214 0.6214
<zzzz> = -21.2052 -0.5696 -0.5696 -0.5696
<xxxy> = 0.6677 -0.6452 -0.6452 -0.6452
<xxxz> = 0.2598 0.5997 0.5997 0.5998
<yyyx> = 0.8732 0.2536 0.2536 0.2536
<yyyz> = -0.2707 -0.2511 -0.2511 -0.2511
<zzzx> = 0.0427 -0.3498 -0.3498 -0.3498
<zzzy> = -0.1990 0.0623 0.0623 0.0623
<xxyy> = -7.1061 -0.8685 -0.8685 -0.8685
<xxzz> = -7.0202 0.3225 0.3225 0.3225
<yyzz> = -6.7823 0.2471 0.2471 0.2471
<xxyz> = -0.0280 0.1887 0.1887 0.1887
<yyxz> = -0.0162 -0.2499 -0.2499 -0.2499
<zzxy> = 0.3612 0.3916 0.3916 0.3916
<r> = 3.9055
<r2> = 3.9949
<r3> = 5.2138
<r4> = 8.1711
Volume = 4.3812
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