Cartesian Multipole Moment Tensors and Polytensors.
 Unabridged Moments
 Symmetric and Compressed Tensors
 Translation of Unabridged Moments
 Rotation of Unabridged Moments
 Tensor Contraction and Traces
 Traceless Moments
 References and Notes
 Links
1.  Unabridged Moments
The unabridged cartesian moment tensor of order n of a charge
distribution ρ(r) is defined by:
where the integral is over the volume containing the charge distribution
and the moment tensor operator r^{(n)} denotes
the tensor product of vector r by itself n times:
For this reason, r^{(n)} is also called the
nth tensor power of the vector r and is some times
indicated as r^{⊗n}. The symbol ⊗
denotes the Kronecker product of two matrices of arbitrary size.[1Kronecker product (definition), 2Zhang, H.; Ding, F.
Journal of Applied Mathematics
2013, Article ID 296185, 8 pages.]
The elements of the tensor r^{(n)} are all the
products of n factors
r_{α1}r_{α2}r_{α3}…r_{αn}
(usually indicated by the index sets
α_{1}α_{2}α_{3}…α_{n}), where each subscript α_{i} denotes Cartesian
axes x, y, and z. The number of the elements of the tensor
r^{(n)} is the number of ntuples (i.e.,
ordered lists of length n) of a set of three elements x,
y, and z and is equal to 3^{n}.
The element ordering of r^{(n)}
is usually chosen to be canonical,[3Applequist, J.
J. Math. Phys. 1983,
24, 736741.]
which means that the first index is changing fastest (i.e., on proceeding
through the array, α_{1} varies through the values 1,2,3 more
rapidly than α_{2}, which varies more rapidly than
α_{3}, etc.). The canonical order is the order in which array
elements are usually stored in a computer memory.
On the other hand, the elements are said to be in anticanonical order when
the first index is changing slowest.
The elements of the tensor r^{(n)} comprise a
column matrix.[4Definition of column
matrix]
To emphasize this fact, the tensor r^{(n)} of
order n may be represented as a tensor of subdivided order
(n,0) or (0,n), which corresponds to a column or row matrix,
respectively, i.e. r^{(n)} ≅
r^{(n,0)} ≅
r^{(0,n)}. In general, any tensor
T^{(m+n)} of order m + n may be represented
as a tensor of subdivided order (m,n), also called a tensor of type
(m,n), T^{(m,n)},
in which m and n are the order indices. The array of
components, T^{(m,n)}_{α1α2α3…αmβ1β2β3…βn},
comprise a rectangular matrix whose rows are indexed by the set
α_{1}α_{2}α_{3}…α_{m}
and whose columns are indexed by the set
β_{1}β_{2}β_{3}…β_{n}.[3Applequist, J.
J. Math. Phys. 1983,
24, 736741.]
For n = 0, the zeroth order moment
operator is the unit operator,
r^{(0)} = 1 (a scalar),
and for n = 1, the first (order)
moment operator is the position vector:
For n = 2, the second (order)
moment operator r^{(2)} is the dyadic (tensor) product
r⊗r:[5Dyadic
product of two vectors of the same dimension]
The first line of Eq. (1.4) shows that the Kronecker product of the column
matrix r^{(1,0)} ≡ r by itself
yields, by definition,[1Kronecker
product] a column matrix r^{(2,0)} whose
elements are in anticanonical order. Such a column matrix, which represents
a tensor of subdivided order (2,0), can be viewed as the vectorization
of the 3×3 quadrupole matrix operator Q, which represents
a tensor of type (1,1). The vectorization is accomplished by
using the “vec” operator, which stacks the columns of a matrix
one underneath the other to form a single vector.[6Magnus, J. R.; Neudecker, H.
The Annals of Statistics 1979, 7, 381394.,
7Henderson, H. V.; Searle, S. R.
Linear and Multilinear Algebra 1981, 9,
271288.] Since matrix Q is symmetric, i.e.
Q = Q^{†}, the equalities of
the second line of Eq. (1.4) follow straightforwardly. In other words,
Eq. (1.4) states that the 3×3 matrix
rr^{†}, obtained by multiplying each
element of r by each element of r^{†}
or, equivalently, by the Kronecker product
r⊗r^{†}, gives directly,
after vectorization, a tensor of type (2,0) with elements in canonical order.
The index notation
(rr^{†})_{ij} =
r_{i}r_{j} = r_{j}r_{i} =
(rr^{†})_{ji},
where both i and j run over 1, 2 and 3 or x, y
and z, emphasizes the fact that r^{(2)} is a
symmetric tensor.
However, because the tensor is symmetric and
rr^{†} =
(rr^{†})^{†},
the component indexes can be safely exchanged to move from canonical to
anticanonical order, and viceversa.
Similarly, for n = 3, the third (order) moment operator is a
3×3×3 tridimensional array, which can be written as:
It is represented by either a 9×3 matrix
r^{(2,1)} or, equivalently, by a 3×9 matrix
r^{(1,2)}, that are tensors of type (2,1) and (1,2),
respectively. Both matrices are stored columnwise in canonical order, so that
they are implicitly vectorized as a r^{(3,0)} column
matrix.
In index notation, the tensor is indicated by
[r^{(3)}]_{ijk}
= r_{i}r_{j}r_{k},
where i, j and k run over 1, 2, and 3,
i.e. over x, y and z.
Due to the symmetry of the tensor, the ordering of the components
can be changed from canonical to anticanonical mode by simply reversing the
indices.
In general, the procedure outlined in Eqs. (1.4) and (1.5) can be used
to build up the nth moment operator:
where r^{(n)} and
r^{(n−1)} are tensors of type (n,0)
and (n−1,0), respectively.
A computer code for the generation of the component labels in canonical
order of the nth order moment operator
r^{(n)} may require n nested
doloops. The Program 1 provides an example of Fortran 77 code for
n = 3.
Program 1.
Fortran 77 code for the generation of the component labels in canonical order of
the 3rd order cartesian moment.
Character*1 R(3) /'x', 'y', 'z'/
Character*3 S(3**3)
ijk = 0
Do k=1,3
Do j=1,3
Do i=1,3
ijk = ijk + 1
S(ijk)(1:1) = R(i)
S(ijk)(2:2) = R(j)
S(ijk)(3:3) = R(k)
end Do
end Do
end Do
Write(*,'(3(7(a3,", "),/),5(a3,", "),a3)') (S(m),m=1,27)

Generalization to any order n is provided by the Fortran 90 code of
Program 2, which implements the Nested Summation Symbol (NSS) operator:
introduced by Carbó and Besalú.[9Carbó, R.; Besalú, E.
J. Math. Chem. 1993, 13, 331342., 10Carbó, R.; Besalú, E.
J. Math. Chem. 1995, 18, 3772.]
The implied ndimensional vectors of Eq. (1.8) are
j = (j_{1}, j_{2}, …,
j_{n}), i = (i_{1},
i_{2}, …, i_{n}), and f
= (f_{1}, f_{2}, …, f_{n}).
The index n is called the dimension of the NSS. Program 2 is a
translation of the NSS into a generalized nested doloop (GNDL) structure,
which is independent of the dimension of the involved nested sums.
Program 2.
Fortran 90 code for the generation of the component labels in canonical order
of a cartesian moment tensor of any order n.
! 
Module NSS
! 
implicit none
private
public CTLbl, PrtOut
contains
! 
Subroutine CTLbl (j,S)
! 
Integer, intent(out) :: j(:)
Character(len=1), intent(out) :: S(:,:)
Character(len=1), dimension(3), parameter :: R = (/'x', 'y', 'z'/)
Integer :: n, k, num
n = size(j)
! initial NSS parameter values
do k=1,n
j(k) = 1
end do
num = 0 ! count the components (num = 3^n)
! Start GNDL procedure
k = n ! the innermost loop corresponds to the index k = n
do while (k > 0)
if (j(k) > 3) then ! index out of range
j(k) = 1 ! reset this level
k = k  1 ! previous level will be activated, if possible
else
! calculational kernel
num = num + 1
do k=1,n
S(nk+1,num) = R(j(k)) ! canonical order
end do
! end of calculational kernel
k = n
end if
if (k > 0) then
j(k) = j(k) + 1 ! step increment
end if
end do
! End of GNDL procedure
End Subroutine CTLbl
! 
Subroutine PrtOut (n,k,S)
! 
Integer, intent(in) :: n, k
Character(len=1), intent(in) :: S(n,k)
Integer, parameter :: rowlength = 72
Integer :: l, m, fulllines, lastlinelength
Character(len=80) :: fmt
m = (rowlength+1)/(n+2)
fulllines = k/m
if (fulllines*m == k) then
fulllines = fulllines  1
lastlinelength = m  1
if (fulllines >= 1) then
write(fmt,'(1h(,i6,1h(,i2,1h(,i2,12ha1,", "),/),,i2,1h(,i2,&
9ha1,", "),,i2,3ha1))') fulllines,m,n,lastlinelength,n,n
else
write(fmt,'(1h(,i2,1h(,i2,9ha1,", "),,i2,3ha1))') &
lastlinelength,n,n
end if
else
lastlinelength = k  fulllines*m  1
if (fulllines >= 1) then
if (lastlinelength >= 1) then
write(fmt,'(1h(,i6,1h(,i2,1h(,i2,12ha1,", "),/),,i2, &
1h(,i2,9ha1,", "),,i2,3ha1))') fulllines,m,n, &
lastlinelength,n,n
else
write(fmt,'(1h(,i6,1h(,i2,1h(,i2,12ha1,", "),/),,i2, &
3ha1))') fulllines,m,n,n
end if
else
write(fmt,'(1h(,i2,1h(,i2,9ha1,", "),,i2,3ha1))') &
lastlinelength,n,n
end if
end if
write(*,fmt) ((S(l,m),l=1,n),m=1,k)
End Subroutine PrtOut
End Module NSS

Cartesian moment tensors of order n are also called
2^{n}pole moments, i.e. mono, di, quadru, octa, and
hexadecapole for n = 0, 1, 2, 3, 4, respectively.
Applequist has set forth a useful organization for electrostatic
properties,[3Applequist, J.
J. Math. Phys. 1983, 24, 736741.] which
is followed in PAMoC.
Applequist defines a Cartesian polytensor of the first order as a sequence of
Cartesian tensors, each arranged in a column.
The firstdegree polytensor is a stacked list of the (column) moment tensors
in increasing order, and will be designated M
The sequence of component tensors, m^{(i)}
is of indefinite length, but PAMoC for practical purposes truncates the
sequence at the fourth order. Since the number of components of the cartesian
moment tensor m^{(n)} of order n is
3^{n}, the number of components of a polytensor which contains
a sequence of cartesian moment tensors up to the nth order is
n
∑
i=0
3^{i} = (3^{n+1} − 1) / 2.
It is useful to relate the indeces
α_{1}α_{2}…α_{n}
of an element entry of a tensor of order n to a singleindex reference
k_{n} which represents the location in memory after the element
of the tensor. For a complete tensor in canonical order it is
and for the corresponding polytensor it is
K = k_{n} +
(3^{n} − 1)/2.
2.  Symmetric and Compressed Tensors
Each component of the moment tensor operator
r^{(n)} is invariant with respect to permutations
of its suffixes, i.e. the tensor is totally symmetric.
Then, suffixes can be ordered according to their type, i.e.
m_{x1…xnxy1…ynyz1…znz},
where n_{x}, n_{y}, and n_{z},
are the number of times x, y, and z occur in the
component index
α_{1}α_{2}…α_{n}.
The n_{i} are called degree indices and satisfy
n_{x} + n_{y}
+ n_{z} = n.
The following concise notation can be used
where only the three degree indices are implied.
The number of distinct components of a symmetric tensor of order
n is ½(n+1)(n+2). A symmetric tensor which
comprises only its distinct components is said to be compressed and
is usually arranged as a column vector with components in canonical order.
Clearly, compression has significance only if n ≥ 2. The canonical
array of index sets of the compressed tensor is derived from
that of the complete tensor by deleting any index set which does not satisfy
the condition
α_{1}≥α_{2}≥…≥α_{n−1}≥α_{n}.
The corresponding Fortran 90 code is reported in Program 3.
Program 3.
Fortran 90 code for the generation of the component labels in canonical order
of a complete and compressed cartesian moment tensor of any order n
(in conjuction with the module in Program 2).
! 
program GNDL
! 
!
! Nested Summation Symbol (NSS): sum over n (j = i, f)
! Translation of the NSS into a Generalized Nested DoLoop (GNDL)
!
! 
Use NSS
Integer, allocatable, dimension(:) :: j
Character(len=1), allocatable, dimension(:,:) :: S
Character(len=3) :: t
Integer :: k, l, m, n
Logical :: ok
if (IArgC() /= 1) then
write(*,'(/," Enter the order n of the moment operator!",/)')
stop
else
call GetArg (1,t)
read(t,*) n
if (n <= 0) then ! the order is out of range
write(*,'(" Error: order",i3," out of range.")') n
stop
end if
end If
allocate ( j(n), S(n,3**n) )
call CTLbl (j,S)
! Output the complete tensor
write(*,'(/," Full Cartesian Tensor of order n =",i2,/, &
" with components in canonical order",/)') n
call PrtOut (n,3**n,S)
! Compressed tensor
k = 0
do m=1,3**n
ok = .true.
do l=1,n1
ok = ok .and. S(l,m) >= S(l+1,m)
end do
if (ok) then
k = k + 1
do l=1,n
S(l,k) = S(l,m)
end do
end if
end do
! Output the compressed tensor
write(*,'(/," Compressed Cartesian Tensor of order n =",i2,/, &
" with components in canonical order",/)') n
call PrtOut (n,(n+1)*(n+2)/2,S)
deallocate ( j, S )
End Program

Since the number of components of the symmetric compressed tensor
m^{(n)} of order n is (n + 1)(n + 2)/2, the number of
components of a polytensor which contains a sequence of symmetric compressed
tensors up to the nth order is
n
∑
i=0
(i + 1)(i + 2)/2 =
(n + 1)(n + 2)(n + 3)/6 . The degree indeces
n_{x}n_{y}n_{z} of a compressed canonical
totally symmetric Cartesian tensor of order n =
n_{x} + n_{y} + n_{z} can be related to a
single index reference j_{n}
which represents the location in memory after the element of the tensor.
Its position in the corresponding polytensor is
J = j_{n} +
n(n + 1) / 2.
The number of components of complete and compressed Cartesian tensors and
polytensors as a function of their order n is reported in Table 2.1.
Table 2.1 −
Number of components of complete and compressed Cartesian tensors and
polytensors as a function of their order n.
n 
complete tensor 3^{n} 
complete polytensor (3^{n+1} − 1)/2 
compressed tensor (n + 1)(n + 2)/2 
compressed polytensor (n + 1)(n + 2)(n + 3)/6 
0  1  1  1  1 
1  3  4  3  4 
2  9  13  6  10 
3  27  40  10  20 
4  81  121  15  35 
The number of times that each of the (n+1)(n+2)/2 distinct
components of a totally symmetric Cartesian tensor,
m^{(n)}, appears in the complete set of
3^{n} components is given by the number of permutations of
n elements with repetitions of n_{x},
n_{y} and n_{z} elements, i.e.
where j spans the compressed canonical array and n_{i}
is the number of times i appears in the component index of the
jth element. Eq. (2.3) defines the elements of the diagonal matrix
g^{(n)}. It is easily verified that
Tr [g^{(n)}] =
3^{n}.
The values of g_{j}^{(n)} are reported in
Table 2.2 for the compressed canonical tensors of order 0 through 4.
The component labels of complete Cartesian tensors of order 0 through 4 are
reported in Table 2.2, using canonical ordering with both component indeces,
α_{1}α_{2}…α_{n},
and degree indeces, n_{x}n_{y}n_{z}.
The components are numbered by the tensor singleindex k_{n},
defined by eq. (1.9) and by the associated polytensor singleindex K.
The singleindeces j_{n} and J of the canonical
compressed tensors are also reported, together with their multiplicities,
defined by eq. (2.3). In addition, the ordering of compressed tensors adopted
in PAMoC is shown. PAMoC reports the components of the quadrupole
moment in the order they are obtained by vectorizing the upper triangular
part of the symmetric quadrupole matrix given by Eq. (1.4). For the
component ordering of the octupole and hexadecapole moments, PAMoC uses
the same convention adopted by the electronic structure modeling package
GAUSSIAN09.[11Gaussian 09, Revision E.01,
Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.;
Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.;
Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.;
Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.;
Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.;
Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.;
Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.;
Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.;
Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.;
Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.;
Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.;
Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.;
Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.;
Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J.
Gaussian, Inc., Wallingford CT, 2009.]
Table 2.2 − Component indeces,
α_{1}α_{2}…α_{n},
and degree indeces, n_{x}n_{y}n_{z}, of the
elements of complete and compressed Cartesian tensors of order 0 through 4,
numbered by tensor and polytensor singleindeces.

complete tensor (canonical order) 

compressed tensor (canonical order, α_{1}≥α_{2}≥…≥α_{n−1}≥α_{n}) 

compressed tensor (PAMoC order) 
tensor order n 
polytensor single index, K 
tensor single index, k_{n} 
component index α_{1}α_{2}…α_{n} 
degree index n_{x}n_{y}n_{z} 

polytensor single index, J 
tensor single index, j_{n} 
g_{in}^{(n)} 

polytensor single index, I 
tensor single index, i_{n} 
g_{in}^{(n)} 
0  1  1   000   1  1  1   1  1  1 
1  2  1  x  100   2  1  1   2  1  1 
 3  2  y  010   3  2  1   3  2  1 
 4  3  z  001   4  3  1   4  3  1 
2  5  1  xx  200   5  1  1   5  1  1 
 6  2  yx  110   6  2  2     
 7  3  zx  101   7  3  2     
 8  4  xy  110       6  2  2 
 9  5  yy  020   8  4  1   7  3  1 
 10  6  zy  011   9  5  2     
 11  7  xz  101       8  4  2 
 12  8  yz  011       9  5  2 
 13  9  zz  002   10  6  1   10  6  1 
3  14  1  xxx  300   11  1  1   11  1  1 
 15  2  yxx  210   12  2  3     
 16  3  zxx  201   13  3  3     
 17  4  xyx  210         
 18  5  yyx  120   14  4  3     
 19  6  zyx  111   15  5  6     
 20  7  xzx  201         
 21  8  yzx  111         
 22  9  zzx  102   16  6  3     
 23  10  xxy  210       15  5  3 
 24  11  yxy  120         
 25  12  zxy  111         
 26  13  xyy  120       14  4  3 
 27  14  yyy  030   17  7  1   12  2  1 
 28  15  zyy  021   18  8  3     
 29  16  xzy  111         
 30  17  yzy  021         
 31  18  zzy  012   19  9  3     
 32  19  xxz  201       16  6  3 
 33  20  yxz  111         
 34  21  zxz  102         
 35  22  xyz  111       20  10  6 
 36  23  yyz  021       19  9  3 
 37  24  zyz  012         
 38  25  xzz  102       17  7  3 
 39  26  yzz  012       18  8  3 
 40  27  zzz  003   20  10  1   13  3  1 
4  41  1  xxxx  400   21  1  1   21  1  1 
 42  2  yxxx  310   22  2  4     
 43  3  zxxx  301   24  3  4     
 44  4  xyxx  310         
 45  5  yyxx  220   24  4  6     
 46  6  zyxx  211   25  5  12     
 47  7  xzxx  301         
 48  8  yzxx  211         
 49  9  zzxx  202   26  6  6     
 50  10  xxyx  310         
 51  11  yxyx  220         
 52  12  zxyx  211         
 53  13  xyyx  220         
 54  14  yyyx  130   27  7  4   26  6  4 
 55  15  zyyx  121   28  8  12     
 56  16  xzyx  211         
 57  17  yzyx  121         
 58  18  zzyx  112   29  9  12     
 59  19  xxzx  301         
 60  20  yxzx  211         
 61  21  zxzx  202         
 62  22  xyzx  211         
 63  23  yyzx  121         
 64  24  zyzx  112         
 65  25  xzzx  202         
 66  26  yzzx  112         
 67  27  zzzx  103   30  10  4   28  8  4 
 68  28  xxxy  310       24  4  4 
 69  29  yxxy  220         
 70  30  zxxy  211         
 71  31  xyxy  220         
 72  32  yyxy  130         
 73  33  zyxy  121         
 74  34  xzxy  211         
 75  35  yzxy  121         
 76  36  zzxy  112       35  15  12 
 77  37  xxyy  220       30  10  6 
 78  38  yxyy  130         
 79  39  zxyy  121         
 80  40  xyyy  130         
 81  41  yyyy  040   31  11  1   22  2  1 
 82  42  zyyy  031   32  12  4     
 83  43  xzyy  121         
 84  44  yzyy  031         
 85  45  zzyy  022   33  13  6     
 86  46  yxzy  121         
 87  47  zxzy  112         
 88  48  xxzy  211         
 89  49  yyzy  031         
 90  50  zyzy  022         
 91  51  xyzy  121         
 92  52  yzzy  022         
 93  53  zzzy  013   34  14  4   29  9  4 
 94  54  xzzy  112         
 95  55  yxxz  211         
 96  56  zxxz  202         
 97  57  xxxz  301       25  5  4 
 98  58  yyxz  121       34  14  12 
 99  59  zyxz  112         
 100  60  xyxz  211         
 101  61  xzxz  202         
 102  62  yzxz  112         
 103  63  zzxz  103         
 104  64  xxyz  211       33  13  12 
 105  65  yxyz  121         
 106  66  zxyz  112         
 107  67  xyyz  121         
 108  68  yyyz  031       27  7  4 
 109  69  zyyz  022         
 110  70  xzyz  112         
 111  71  yzyz  022         
 112  72  zzyz  013         
 113  73  xxzz  202       31  11  6 
 114  74  yxzz  112         
 115  75  zxzz  103         
 116  76  xyzz  112         
 117  77  yyzz  022       32  12  6 
 118  78  zyzz  013         
 119  79  xzzz  103         
 120  80  yzzz  013         
 121  81  zzzz  004    35  15  1  23  3  1 
From an algebraic point of view, a (n+1)(n+2)/2 ×
3^{n} matrix C can be defined to transform a complete
canonical tensor, m^{(n)}, into a
compressed canonical tensor,
m^{(n)}:
with the inverse transformation:[20
The product C^{†} C is not positive definite and
cannot be inverted.
In addition, C^{†}
g^{(n)} C ≠ I.]
The compression matrix C is defined in such a way:
i.e. let the rowsuffix k of a generic element C_{kl}
specify the canonical sequence number of the compressedtensor component,
whereas the columnsuffix l is the canonical sequence number of a
completetensor component. Then C_{kl} =
g_{k}^{−1}
for all values of l which identify any of the g_{k}
components of the complete tensor equal by symmetry to the kth
component of the compressed tensor. Otherwise, C_{kl} = 0.
However, it is worth noting that PAMoC never uses matrix C,
explicitly.
3.  Translation of Unabridged Moments
Consider the transformation of the unabridged cartesian moment tensor
operator of order 1 (position vector) under translation of the axis system
given by
In eq. (3.1), r
is a point in the translated axis system. r is the same point
in the original axis system. R =
 X Y Z ^{†} is the
position vector of the new origin in the original axis system.
Combining eqs (1.1) and (3.1) yields the transformation of the dipole moment
under translation of the axis system
Similarly, the transformation of the unabridged cartesian moment tensor
operators of order 2 through 4 under translation of the axis system is given
by
or, in index notation:
Substituting expressions (3.4) for the moment operator in Eq. (1.1) and
integrating yields:
The results above can be put together into a set of coupled equations, that
in matrix notation have the form:
where M is the polytensor defined by Eq. (3.1), which contains the
original multipole moments, M
contains the translated multipole moments and the shift matrix
S is a lower unitriangular matrix of dimension N×N,
with N =
(n+1)(n+2)(n+3)/6, which is factorized as shown in
Eq. (3.7):
where the
I_{(n+1)(n+2)/2}
are identity matrices, the
0_{[(n+1)(n+2)/2]×[N−(n+1)(n+2)(n+3)/6]}
are rectangular matrices whose elements are equal to zero, and the elements of
the rectangular matrices
S_{[(n+1)(n+2)/2]×[n(n+1)(n+2)/6]}
are simple functions of the elements of R:
Monopoles, m^{(0)}, do not change under translation of
the axis system, because they are scalars. Eq. (3.1) shows that also
the components of the dipole remain unchanged under translation of the axis
system provided that the monopole is null. In general, according to
Eqs. (3.5), a symmetric Cartesian tensor m^{(n)}
of order n is invariant under translation of the axis system
provided that all the symmetric Cartesian tensors of lower order,
up to n−1, are null.
On the other hand, the translation of a polytensor,
whose component tensors are all null but the monopole m^{(0)},
generates a polytensor with nonnull component tensors of any order, i.e.
m^{(1)} = −Rm^{(0)},
m^{(2)} = X^{2} XY Y^{2} XZ YZ Z^{2}^{†} m^{(0)},
m^{(3)} = − X^{3} Y^{3} Z^{3} XY^{2} YX^{2} ZX^{2} XZ^{2} YZ^{2} ZY^{2} XYZ^{†} m^{(0)},
etc.
4.  Rotation of Unabridged Moments
In general, a tensor of order n is a mathematical object with
n suffixes, T_{ijk…}, which obeys the
transformation law
where L is the rotation matrix between frames. A scalar,
T, is a tensor of order 0, because it is the same in all frames
(T' = T). Any vector, like the position vector r, is
a tensor of order 1, because r'_{i} =
∑_{p} L_{ip} r_{p},
or in matrix notation r' =
L r. For second order tensors, such as the
quadrupole tensor, the transformation law
T'_{ij} = ∑_{pq}
L_{ip}L_{jq} T_{pq}, can be
rewritten in matrix notation as T' =
L T L^{†}, in addition to the tensor
notation T'^{(2)} =
L^{(2,2)} T^{(2)}.
In the previous sections we defined multidimensional arrays as tensor. Here
we define tensors through the transformation low that they obey, as in
Eq. (4.1). Definition (4.1) is based on a synecdoche, since the entire tensor
is indicated by one of its components (the so called suffix notation).
In the present development, the common convention of implied summation over
repeated indices is not being used. If we did, we would not bother to write
down the ∑ in the right side of Eq. (4.1) because suffixes p,
q and r appear twice in a single term of an expression.
As an example, let's write Eq. (4.1) for the second order Cartesian moment
m^{(2)}, using the suffix notation:
where the two sums in the second member have been developed following
the algorithm of Program 1, so that the nine tensor components appear in
canonical order in the sum of the last member.
Since Cartesian tensors, m^{(n)}, are totally
symmetric, there are only (n+1)(n+2)/2 distinct components
(namely, six for n = 2). The number of times that each of the distinct
components appears in the complete set of 3^{n} components
is given by Eq. (2.2). Then, Eq. (4.2) can be rewritten as
where k and l span the compressed canonical array
which comprises (n+1)(n+2)/2 components; {pq} spans
the g_{k} doublets (p,q) which have
m_{k}^{(2)} as a common factor, and (i,j)
is the doublet of suffixes which corresponds to the lth suffix
of the compressed canonical array.
The last member of Eq. (4.2) defines, by elements, the rotation matrix
L^{(2,2)} of the complete canonical tensor
m^{(2)}:
On the other hand, the rotation matrix of the corresponding compressed
canonical tensor is defined by the last member of Eq. (4.3):
where the diagonal matrix g^{(n)} and the
rectangular matrix C are defined by Eqs (2.25).
It's worth noting that, given the rotated position vector r' = L r,
the primed moment operator of order 2, r'^{(2)},
according to Eq. (1.4), is given by the Kronecker product
r'⊗r', which leads to:
where the mixedproduct property has been applied.[2Zhang, H.; Ding, F.
Journal of Applied
Mathematics 2013, Article ID 296185, 8 pages.]
Eq. (4.6) can be generalized to r'^{(n)} =
L^{(n,n)}r^{(n)}
and integrated by Eq. (1.1), yielding
5.  Tensor Contraction and Traces
The contraction of a tensor is obtained by setting unlike indices equal
and summing. Contraction reduces the tensor order by 2. Contraction is not
defined for tensors of order 0 (scalars) and order 1 (vectors). For a
second order tensor m^{(2)}, which has two indices,
the contraction is the same as the trace (provided that
m^{(2)} is interpreted as a matrix) and is therefore
a scalar:
For a totally symmetric third order cartesian tensor
m^{(3)}, which has three indices, contraction implies
three trace relationships, each one defining a component of a first order
tensor or vector, m^{(3:1)}:
Finally, for a totally symmetric fourth order cartesian tensor
m^{(4)}, which has four indices, contraction implies
six distinct trace relationships, each one defining a component of a
symmetric compressed second order tensor, m^{(4:1)}:
The double contraction of a tensor is obtained by setting four unlike
indices two by two to be equal and summing over the two pairs of equal indices.
Double contraction reduces the tensor order by 4. Therefore, double
contraction of a totally symmetric fourth order cartesian tensor
m^{(4)} yields a scalar:
In general,[12Applequist, Jon
Chem. Phys. 1984, 85, 279290.;
13Applequist, Jon
J. Chem. Phys.
1985, 83, 809826.;
14Applequist, Jon
J. Phys. A: Math. Gen. 1989, 22, 43034330.]
the trace of a tensor m^{(n)} of order n
with respect to one pair of suffixes is a tensor
m^{(n:1)} of order n−2 and is given
by elements:
where k_{x} + k_{y} + k_{z} =
n − 2.
The distinct components of m^{(n:1)} are
exactly n(n−1)/2, one for each selection of the
(n−2) suffixes x, y and z. The
trace of a tensor m^{(n)} of order n
with respect to l pairs of suffixes (called an “lfold
trace”) is a tensor
m^{(n:l)} of order (n −
2l) and is given by elements:[14Applequist, Jon
J. Phys. A: Math. Gen.
1989, 22, 43034330.]
where k_{x} + k_{y} + k_{z} =
n − 2l and the sum is over all nonnegative indices such
that l_{x} + l_{y} + l_{z} =
l. The trinomial coefficient g(l;l_{x}l_{y}l_{z}) = l!/l_{x}!l_{y}!l_{z}!
appears here as the number of the ways one can place l_{x}
pairs xx, l_{y} pairs yy, and l_{z}
pairs zz in the indices r_{1}r_{1}
r_{2}r_{2} …
r_{l}r_{l} of the lfold
trace.[14Applequist, Jon
J. Phys. A: Math. Gen. 1989, 22, 43034330.]
For instance, if l = 2 there are six distinct sets of indices
l_{x}l_{y}l_{z} (namely, 200, 020,
002, 110, 101, 011, with multiplicity 1, 1, 1, 2, 2, 2, respectively) so that
expression (5.4) is easily recovered. If l = 1 there are three distinct
sets of indices, i.e. 100, 010, 001, so that eqs (5.4), (5.3) and (5.2) are
obtained.
If the trace vanishes regardless of which index pair is contracted,
the tensor is said to be totally traceless.
If a tensor is totally symmetric and traceless in one index pair, then it is
traceless for all index pairs, and is said to be totally symmetric and
traceless.
6.  Traceless Moments
The traceless cartesian moment tensor of order n of a charge
distribution ρ(r) is defined by elements in
component index notation[15Stone, A. J.
"The Theory of Intermolecular
Forces"
Oxford University Press, Oxford, 2000;
p. 16.; 16Coppens, P.
"XRay Charge Densities and Chemical Bonding"
International Union of
Crystallography, Oxford University Press, Oxford, 1997; p. 144.]
following the convenction that r_{i} subscripts denote
cartesian axes x, y, z, or by elements in degree index
notation[14Applequist, Jon
J. Phys. A: Math. Gen. 1989, 22, 43034330.]
where n_{i} is the number of times i occurs in the
index set
r_{1}r_{2}…r_{n−1}r_{n}
and n_{x} + n_{y} + n_{z} =
n. The latter notation is usefull for compressed tensors, as it addresses
only distinct components of the complete tensors.
Direct differentiation of r^{−1} yields in index
notation
where ⌊n/2⌋ denotes the integer part of n/2;
(2n − 2m − 1)!! = (2n −
2m)!/2^{n−m} (n − m)! =
1 ⋅ 3 ⋅ 5 ⋅ (2n − 2m − 1)
with (−1)!! = 1; δ_{ij} is the Kronecker
delta; and the sum over T{r_{i}} is the
sum over all permutations of the symbols r_{1}
r_{2} … r_{n} which
give distinct terms.
Inserting the moment operators of Eq. (7.3) into Eq. (7.1) and
integrating yields:[12Applequist, Jon
Chem. Phys.
1984, 85, 279290.]
The order n of tensors m has been omitted in
Eqs (7.912) because its value is clear from the number of suffixes. The symbols
μ, Θ, Ω, and
Φ, have been adopted [17Buckingham, A. D.
J. Chem. Phys. 1959,
30, 15801585.; 18Buckingham, A. D.
Advan. Chem. Phys.
1967, 12, 107142.; 19McLean, A. D.; Yoshimine, M. J. Chem. Phys.
1967, 47, 19271935.] for
μ^{(1)}, μ^{(2)},
μ^{(3)}, and μ^{(4)},
respectively. In addition:
According to Eq. (5.6), the elements of tensor
r^{(n:m)}, which is the trace of
r^{(n)} with respect to m pairs of
suffixes (called an “mfold trace”), can be written
explicitly as
where k_{x} + k_{y} + k_{z} =
n − 2m. In addition[21Trinomial expansion and trinomial coefficients]
so that
where the sum is over all nonnegative indices m_{x},
m_{y}, m_{z} such that m_{x} +
m_{y} + m_{z} = m, and
2m_{i} ≤ k_{i} ∀ i =
x,y,z, and k_{x} +
k_{y} + k_{z} = n − 2m.
[Please, note that in the equations above r_{i} denotes any
cartesian coordinate x, y, z, whereas r without
suffix denotes r = (x^{2} + y^{2}
+ z^{2})^{−½}].
In tensor notation, equations (7.1) and (7.2) become:
where the nabla symbol ∇ denotes the vector differential
operator:
and ∇^{2} denotes the Hessian matrix operator
whose trace is the laplacian operator, ∇^{2}:
The traceless condition with respect to any pair of suffixes comes from the fact that
the laplacian of r^{−1} is null:[15Stone, A. J.
"The Theory of Intermolecular Forces"
Oxford University Press, Oxford, 2000; p. 16.]
The last equation of system (7.3) shows that the nth gradient of
r^{−1} is a linear combination of
r^{(n)} and its traces in all pairs of indices.[12Applequist, Jon
Chem. Phys. 1984, 85, 279290.]
In other words, the totally symmetric tensor
r^{(n)} of order n can be transformed
in a traceless totally symmetric tensor
∇^{(n)}r^{−1} by the action
of the detracer operator 𝔇_{n},[12Applequist, Jon
Chem. Phys. 1984,
85, 279290.; 13Applequist, Jon
J. Chem. Phys. 1985,
83, 809826.] which linearly combines the
tensor components through the associated square matrix
D^{(n,n)}:[14Applequist, Jon
J. Phys. A: Math. Gen.
1989, 22, 43034330.]
or in index notation
Inserting Eq. (7.5) into Eq. (7.1) and integrating, yields:
From (7.3) and (7.4) it can be seen that the components of the tensor of
subdivided order D^{(n,n)} [3Applequist, J.
J. Math. Phys. 1983,
24, 736741.] have the following expression:[14Applequist, Jon
J. Phys. A: Math. Gen.
1989, 22, 43034330.]
where the sum over T{αβ} is the
sum over all permutations of the symbols α_{1}
α_{2} … α_{n} and
β_{1} β_{2} …
β_{n} giving distinct terms.
For example, Eq. (7.8) gives for n = 2
which, using Eq. (7.6), gives the third equation of system (7.3).
Table − Using the last equation
of system (7.3) to find the first four equations of the same system.
α_{1}
α_{2} …
α_{n} 
n 
m 
(−1)^{m}(2n
− 2m − 1)!! 
∑
T{α} … 
α  1  0  −1 
r_{α} 
α_{1}α_{2} 
2  0  3 
r_{α1}
r_{α2} 
  1  −1 
δ_{α1α2} 
α_{1}α_{2}α_{3} 
3  0  15 
r_{α1}
r_{α2}
r_{α3} 
  1  −3 
δ_{α1α2} r_{α3} +
δ_{α1α3} r_{α2} +
δ_{α2α3} r_{α1} 
α_{1}α_{2}α_{3}α_{4} 
4  0  105 
r_{α1}
r_{α2}
r_{α3}
r_{α4} 
  1  −15 
δ_{α1α2} r_{α3}r_{α4} +
δ_{α1α3} r_{α2}r_{α4} +
δ_{α1α4} r_{α2}r_{α3} +
δ_{α2α3}
r_{α1}r_{α4} +
δ_{α2α4}
r_{α1}r_{α3} +
δ_{α3α4}
r_{α1}r_{α2} 
 
2  3 
δ_{α1α2} δ_{α3α4} +
δ_{α1α3} δ_{α2α4} +
δ_{α1α4} δ_{α2α3} 
References and Notes
 If A is an m × n matrix and B
is a p×q matrix, then the Kronecker product
A⊗B is the mp×nq block matrix:
See reference 2 and external links L7
and L8 for more details.
 "On the Kronecker Products and Their Applications"
Zhang, H.; Ding, F. Journal of Applied Mathematics
2013, Article ID 296185, 8 pages,
http://dx.doi.org/10.1155/2013/296185
 "Cartesian polytensors"
Applequist, J. J. Math. Phys. 1983, 24, 736741.
 A matrix which has only one column is called a column matrix.
The transpose of a column matrix is another matrix which has only one
row and is called a row matrix. Since a matrix is defined as a
bidimensional array whose elements have exactly two indices, a column
matrix has dimensions n×1, and a row matrix has
dimensions 1×n. In spite of this, they are often indicated
as vectors, which instead are defined as monodimensional arrays of
elements with exactly one index.
Programming languages like Fortran that support multidimensional
arrays typically have a native columnmajor or canonical storage order
for these arrays, that means that consecutive elements of the columns of
the arrays are contigous in memory. This linear storage of multidimensional
arrays or tensors complies with tensors being represented by a column
matrix.
 A dyadic product is the special case of the Kronecker
product or tensor product between two vectors of the same dimension.
If a = a_{x}
a_{y} a_{z}^{†}
and b = b_{x}
b_{y} b_{z}^{†} are two
cartesian vectors, their dyadic products are:
and, similarly:
The two equations above show that the dyadic product is not commutative.
The tensor product of two vectors of different dimensions is called
outer product [cf Link 6].
It contrasts with the inner product of vectors a
and b, which yields a scalar:
and is better known as the dot product.
The inner product is the trace of the outer product.
 "The commutation matrix: some properties and applications"
Magnus, J. R.; Neudecker, H. The Annals of Statistics 1979,
7, 381394.
 "The VecPermutation Matrix, The Vec Operator and Kronecker
Products: A Review"
Henderson, H. V.; Searle, S. R.
Linear and Multilinear Algebra 1981, 9, 271288.
 In general, a tensor r^{(2,0)} with
components in anticanonical order can be transformed into an
equivalent tensor with components in canonical order by means of
a suitable vecpermutation matrix,[7Henderson, H. V.; Searle, S. R.
Linear and Multilinear Algebra 1981, 9,
271288.] also known as commutation matrix:[6Magnus, J. R.; Neudecker, H.
The Annals of Statistics 1979, 7, 381394.]
 "Nested summation symbols and perturbation theory"
Carbó, R.; Besalú, E. J. Math. Chem.
1993, 13, 331342.
 "Definition and quantum chemical applications of nested
summation symbols and logical functions: Pedagogical artificial
intelligence devices for formulae writing, sequential programming
and automatic parallel implementation"
Carbó, R.; Besalú, E. J. Math. Chem.
1995, 18, 3772.
 Gaussian 09, Revision E.01, Frisch, M. J.; Trucks, G. W.;
Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.;
Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.;
Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.;
Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.;
Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.;
Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.;
Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.;
Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.;
Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.;
Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.;
Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.;
Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.;
Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.;
Dapprich, S.; Daniels, A. D.; Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.;
Cioslowski, J.; Fox, D. J. Gaussian, Inc., Wallingford CT, 2009.
 "Fundamental relationships in the theory of electric multipole
moments and multipole polarizabilities in static fields"
Applequist, Jon Chem. Phys. 1984, 85, 279290.
 "A multipole interaction theory of electric polarization of
atomic and molecular assemblies"
Applequist, Jon J. Chem. Phys. 1985, 83, 809826.
 "Traceless Cartesian tensor forms for spherical harmonic
functions: new theorems and applications to electrostatics of
dielectric media"
Applequist, Jon J. Phys. A: Math. Gen. 1989, 22,
43034330.
 Stone, A. J. "The Theory of Intermolecular Forces",
Oxford University Press, Oxford, 2000, ISBN 019855883X; p. 16.
 Coppens, P. "XRay Charge Densities and Chemical Bonding",
International Union of Crystallography, Oxford University Press, Oxford,
1997, ISBN: 9780195098235; p. 144.
 "Direct Method of Measuring Molecular Quadrupole Moments"
Buckingham, A. D. J. Chem. Phys. 1959, 30,
15801585.
 "Permanent and Induced Molecular Moments and LongRange
Intermolecular Forces"
Buckingham, A. D. Advan. Chem. Phys. 1967, 12,
107142.
 "Theory of Molecular Polarizabilities"
McLean, A. D.; Yoshimine, M. J. Chem. Phys. 1967, 47,
19271935.
 The product C^{†} C is not positive
definite and cannot be inverted. In addition, C^{†}
g^{(n)} C ≠ I.
 The nth power of a trinomial a + b + c can be
expanded into a sum of (n+1)(n+2)/2 monomials according to
the following equation
where i, j, k are nonnegative integers and
are the trinomial coefficients. Their sum over all nonnegative
indices i, j, k such that i + j + k = n equals
3^{n}:
A trinomial coefficient can be expressed as the product of three binomial
coefficients:
This identity can be verified by applying the definition of binomial
coefficient, i.e.
(
n
i
) =
n!
/
i! (n−i)!.
It's worth noting that the binomial coefficient
(
n
i
) can also be
written as
(
n
i,j
) =
n!
/
i! j!, provided
that j = n − i.
Links
 Rowland, T.;
Weisstein, E. W. "Tensor." From MathWorld  A Wolfram Web
Resource.
 "Tensor."
Wikipedia^{®}, the free encyclopedia.

"Cartesian tensor."
Wikipedia^{®}, the free encyclopedia.
 "Tensor
product." Wikipedia^{®}, the free encyclopedia.
 Rowland, T.
"Vector Space Tensor Product." From MathWorld  A Wolfram Web Resource.
 "Outer
product." Wikipedia^{®}, the free encyclopedia.
 "Kronecker
product." Wikipedia^{®}, the free encyclopedia.
 Weisstein, E. W.
"Kronecker Product." From MathWorld  A Wolfram Web Resource.

"Vectorization of a matrix." Wikipedia^{®}, the free encyclopedia.