Cartesian Multipole Moment Tensors and Polytensors.

1. Unabridged Moments
2. Symmetric and Compressed Tensors
3. Translation of Unabridged Moments
4. Rotation of Unabridged Moments
5. Tensor Contraction and Traces
6. Traceless Moments
7. References and Notes
8. 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 n-th 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_α1r_α2r_α3…r_αn (usually indicated by the index sets α₁α₂α₃…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 n-tuples (i.e., ordered lists of length n) of a set of three elements x, y, and z and is equal to 3ⁿ. The element ordering of r⁽ⁿ⁾ is usually chosen to be canonical,[3Applequist, J.
J. Math. Phys. 1983, 24, 736-741.] which means that the first index is changing fastest (i.e., on proceeding through the array, α₁ varies through the values 1,2,3 more rapidly than α₂, which varies more rapidly than α₃, 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 α₁α₂α₃…α_m and whose columns are indexed by the set β₁β₂β₃…β_n.[3Applequist, J.
J. Math. Phys. 1983, 24, 736-741.]

For n = 0, the zero-th 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, 381-394., 7Henderson, H. V.; Searle, S. R.
Linear and Multilinear Algebra 1981, 9, 271-288.] 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_ir_j = r_jr_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_ir_jr_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 n-th 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 n-th order moment operator r⁽ⁿ⁾ may require n nested do-loops. The Program 1 provides an example of Fortran 77 code for n = 3.

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, 331-342., 10Carbó, R.; Besalú, E.
J. Math. Chem. 1995, 18, 37-72.] The implied n-dimensional vectors of Eq. (1.8) are j = (j₁, j₂, …, j_n), i = (i₁, i₂, …, i_n), and f = (f₁, f₂, …, f_n). The index n is called the dimension of the NSS. Program 2 is a translation of the NSS into a generalized nested do-loop (GNDL) structure, which is independent of the dimension of the involved nested sums.

!     ------------------------------------------------------------------
      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(n-k+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 2ⁿ-pole moments, i.e. mono-, di-, quadru-, octa-, and hexadeca-pole 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, 736-741.] 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 first-degree 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 3ⁿ, the number of components of a polytensor which contains a sequence of cartesian moment tensors up to the n-th order is n ∑ i=0 3ⁱ = (3ⁿ⁺¹ − 1) / 2. It is useful to relate the indeces α₁α₂…α_n of an element entry of a tensor of order n to a single-index 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 + (3ⁿ − 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 α₁α₂…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   α₁≥α₂≥…≥α_n−1≥α_n. The corresponding Fortran 90 code is reported in Program 3.

!     ------------------------------------------------------------------
      program GNDL
!     ------------------------------------------------------------------
!
!     Nested Summation Symbol (NSS): sum over n (j = i, f)
!     Translation of the NSS into a Generalized Nested Do-Loop (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,n-1
            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 n-th order is    n ∑ i=0 (i + 1)(i + 2)/2 = (n + 1)(n + 2)(n + 3)/6 .   The degree indeces n_xn_yn_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.

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 3ⁿ 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 j-th element. Eq. (2.3) defines the elements of the diagonal matrix g⁽ⁿ⁾. It is easily verified that Tr [g⁽ⁿ⁾] = 3ⁿ. 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, α₁α₂…α_n, and degree indeces, n_xn_yn_z. The components are numbered by the tensor single-index k_n, defined by eq. (1.9) and by the associated polytensor single-index K. The single-indeces 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 GAUSSIAN-09.[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.]

From an algebraic point of view, a (n+1)(n+2)/2 × 3ⁿ 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 row-suffix k of a generic element C_kl specify the canonical sequence number of the compressed-tensor component, whereas the column-suffix l is the canonical sequence number of a complete-tensor 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 k-th 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 non-null component tensors of any order, i.e.
m⁽¹⁾ = −Rm⁽⁰⁾,
m⁽²⁾ = |X²  XY  Y²  XZ  YZ  Z²|^† m⁽⁰⁾,
m⁽³⁾ = − |X³  Y³  Z³  XY²  YX²  ZX²  XZ²  YZ²  ZY²  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_ipL_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 3ⁿ 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 l-th 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 (2.2-5).

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 mixed-product 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, 279-290.; 13Applequist, Jon
J. Chem. Phys. 1985, 83, 809-826.; 14Applequist, Jon
J. Phys. A: Math. Gen. 1989, 22, 4303-4330.] 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 “l-fold 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, 4303-4330.]

where k_x + k_y + k_z = n − 2l and the sum is over all non-negative indices such that l_x + l_y + l_z = l. The trinomial coefficient g(l;l_xl_yl_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 r₁r₁ r₂r₂ … r_lr_l of the l-fold trace.[14Applequist, Jon
J. Phys. A: Math. Gen. 1989, 22, 4303-4330.] For instance, if l = 2 there are six distinct sets of indices l_xl_yl_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.
"X-Ray 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, 4303-4330.]

where n_i is the number of times i occurs in the index set r₁r₂…r_n−1r_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 r₁ r₂ … 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, 279-290.]

The order n of tensors m has been omitted in Eqs (7.9-12) 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, 1580-1585.; 18Buckingham, A. D.
Advan. Chem. Phys. 1967, 12, 107-142.; 19McLean, A. D.; Yoshimine, M. J. Chem. Phys. 1967, 47, 1927-1935.] 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 “m-fold 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 non-negative 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| = (x² + y² + z²)^−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 n-th 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, 279-290.] 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, 279-290.; 13Applequist, Jon
J. Chem. Phys. 1985, 83, 809-826.] which linearly combines the tensor components through the associated square matrix D^(n,n):[14Applequist, Jon
J. Phys. A: Math. Gen. 1989, 22, 4303-4330.]

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, 736-741.] have the following expression:[14Applequist, Jon
J. Phys. A: Math. Gen. 1989, 22, 4303-4330.]

where the sum over T{αβ} is the sum over all permutations of the symbols α₁ α₂ … α_n and β₁ β₂ … 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).