3.8.4. horton.matrix.dense
– Dense matrix implementations¶
3.8.4.1. Naming scheme for the contract, slice or expand methods¶
The name of contract
, slice
and expand
methods is as follows:
[iadd_]{contract,slice,expand}[_X][_Y][_to_Z]
where each part between square brackets is optional. X
, Y
and Z
can be any of one
, two
, three
or four
. The names
contract
, slice
and expand
have the following meaning:
contract
 Some actual contraction takes place, i.e. summing of (products) of matrix elements.
slice
 A subset of the elements is merely selected, without any addition of
elements. This is never combined with
iadd_
. expand
 Products of elements are computed but these products are not added. Similar to an outer product but more general.
When iadd_
is used as a prefix, the result of the contraction is added
inplace to the self object. In this case, the _to_Z
part is never
present. A contraction of all input arguments is made. The dimensionality
of the input arguments is indicated by _X
and _Y
.
When _to_Z
is present, the contraction involves self and possibly other
arguments whose dimensionality is indicated by _X
and _Y
. In this
case, iadd_
can not be present. The result is written to an output
argument. If the output argument is not provided, fresh memory is allocated
to compute the contraction and the result is returned. (This allocation is
provided for convenience but should not be used in critical situations.)
Some remarks:
 Similar conventions apply to an
expand
method.  All
contract
andexpand
methods are implemented with the driver methodDenseLinalgFactory.einsum
. However, other implementations than Dense are free to implement things differently.  All
contract
andexpand
methods never touch the internals of higherindex objects.
For more specific information, read the documentation of the individual classes and methods below.
3.8.4.2. Handling of index symmetry¶
The dense matrix classes do not exploit matrix symmetry to reduce memory
needs. Instead they will happily store nonsymmetric data if need be. There
are however a few methods in the DenseTwoIndex
and
DenseFourIndex
classes below that take a symmetry
argument to
check or enforce a certain index symmetry.
The symmetry argument is always an integer that corresponds to the redundancy of the offdiagonal matrix elements in the dense storage. In practice this means the following:
DenseTwoIndex
symmetry=1
: Nothing is checked/enforcedsymmetry=2
: Hermitian index symmetry is checked/enforced (default), i.e. \(\langle i \vert A \vert j \rangle = ` :math:\)langle j vert A vert i rangle`
DenseFourIndex
symmetry=1
: Nothing is checked/enforcedsymmetry=2
: Dummy index symmetry is checked/enforced, i.e. \(\langle ij \vert B \vert kl \rangle =\) \(\langle ji \vert B \vert lk \rangle\)symmetry=4
: Hermitian and real index symmetry are checked/enforced, i.e. \(\langle ij \vert B \vert kl \rangle =\) \(\langle kl \vert B \vert ij \rangle =\) \(\langle kj \vert B \vert il \rangle =\) \(\langle il \vert B \vert kj \rangle\). (This only makes sense because the basis functions are assumed to be real.)symmetry=8
: All possible symmetries are checked/enforced, i.e. \(\langle ij \vert B \vert kl \rangle =\) \(\langle kl \vert B \vert ij \rangle =\) \(\langle kj \vert B \vert il \rangle =\) \(\langle il \vert B \vert kj \rangle =\) \(\langle ji \vert B \vert lk \rangle =\) \(\langle lk \vert B \vert ji \rangle =\) \(\langle jk \vert B \vert li \rangle =\) \(\langle li \vert B \vert jk \rangle\). (This only makes sense because the basis functions are assumed to be real.)
3.8.4.3. Dense matrix classes¶

class
horton.matrix.dense.
DenseLinalgFactory
(default_nbasis=None)¶ Bases:
horton.matrix.base.LinalgFactory
Optional arguments:
 default_nbasis
 The default basis size when constructing new operators/expansions.

create_expansion
(nbasis=None, nfn=None)¶ Create a DenseExpansion with defaults from the LinalgFactory
Optional arguments:
 nbasis
 The number of basis functions. When not given, the default_nbasis value of the DenseLinalgFactory instance will be used.
 nfn
 The number of orbitals. When not given, the default_nbasis value of the DenseLinalgFactory instance will be used.

create_four_index
(nbasis=None, nbasis1=None, nbasis2=None, nbasis3=None)¶ Create a DenseFourIndex with defaults from the LinalgFactory
Optional arguments:
 nbasis
 The number of basis functions. When not given, the default_nbasis value of the DenseLinalgFactory instance will be used.

create_one_index
(nbasis=None)¶ Create a DenseOneIndex with defaults from the LinalgFactory
Optional arguments:
 nbasis
 The number of basis functions. When not given, the default_nbasis value of the DenseLinalgFactory instance will be used.

create_three_index
(nbasis=None, nbasis1=None, nbasis2=None)¶ Create a DenseThreeIndex with defaults from the LinalgFactory
Optional arguments:
 nbasis
 The number of basis functions. When not given, the default_nbasis value of the DenseLinalgFactory instance will be used.

create_two_index
(nbasis=None, nbasis1=None)¶ Create a DenseTwoIndex with defaults from the LinalgFactory
Optional arguments:
 nbasis
 The number of basis functions. When not given, the default_nbasis value of the DenseLinalgFactory instance will be used.
 nbasis1
 The number of basis functions for the second axis if it differes
from
nbasis
.

static
einsum
(subscripts, out, factor, clear, *operands)¶ einsum wrapper for DenseNBody objects
Arguments:
 subscripts
 The usual subscripts argument for the einsum function, except
that the format is slightly limited. All indexes must be
explicitly stated. If the output argument is a scalar,
>
may not be present. In all other cases,>
must be present.  out
 An output DenseNBody object. This argument is not allowed when the result is a scalar.
 factor
 Multiply the contraction by a scalar factor.
 clear
 When set to False, the output argument is not zeroed before adding the contraction.
 operands
 A list where each element is (i) a DenseNBody object or (ii) a twotuple of DenseNBody object and ranges to select for slicing. The ranges must be a tuple of integers of the form (begin0, end0, begin1, end1, ...) where the number of (begin, end) pairs depends on the dimensionality of the corresponding tensor.
Returns: the out object. When the out argument is given, this is returned with inplace modifications.

from_hdf5
(grp)¶ Construct an instance from data previously stored in an h5py.Group.
Arguments:
 grp
 An h5py.Group object.

set_default_nbasis
(nbasis)¶

static
tensordot
(a, b, axes, out=None, factor=1.0, clear=True)¶ tensordot wrapper for dense nindex objects.
Arguments:
 a, b, axes
 See documentation of numpy.tensordot
Optional arguments:
 out
 The output argument, an instance of one of the Dense?Index classes.

to_hdf5
(grp)¶ Write a LinalgFactory to an HDF5 group
Argument:
 grp
 A h5py.Group instance to write to.

class
horton.matrix.dense.
DenseOneIndex
(nbasis)¶ Bases:
horton.matrix.base.OneIndex
Arguments:
 nbasis
 The number of basis functions.

assign
(other, begin0=0, end0=None)¶ Assign with the contents of another object
Arguments:
 other
 Another DenseOneIndex object or a scalar.
Optional arguments:
 begin0, end0
 If specified, only a sublock of self is assigned.

change_basis_signs
(signs)¶ Correct for different sign conventions of the basis functions.
Arguments:
 signs
 A numpy array with sign changes indicated by +1 and 1.

clear
()¶ Reset all elements to zero.

copy
(begin=0, end=None)¶ Return a copy of (a part of) the object
Optional arguments:
 begin, end
 Can be used to select a subblock of the object. When not given, the full range is used.

divide
(other, factor=1.0, out=None)¶ Divide two DenseOneIndex object, multiplied by factor, and return output
Arguments:
 other
 A DenseOneIndex instance to be divided
Optional arguments:
 factor
 A scalar factor
 out
 The output DenseOneIndex

dot
(other, factor=1.0)¶ Dot product with other DenseOneIndex object, multiplied by factor
Arguments:
 other
 A DenseOneIndex instance to be added
Optional arguments:
 factor
 A scalar factor

classmethod
from_hdf5
(grp)¶ Construct an instance from data previously stored in an h5py.Group.
Arguments:
 grp
 An h5py.Group object.

get_element
(i)¶ Return a matrix element

get_max
(abs=True)¶ Return maximum (absolute) element

iadd
(other, factor=1.0)¶ Add another DenseOneIndex object inplace, multiplied by factor
Arguments:
 other
 A DenseOneIndex instance to be added
Optional arguments:
 factor
 A scalar factor

iscale
(factor)¶ Inplace multiplication with a scalar
Arguments:
 factor
 A scalar factor.

mult
(other, out=None, factor=1.0)¶ Muliply with other DenseOneIndex object, multiplied by factor
Arguments:
 other
 A DenseOneIndex instance to be added
Optional arguments:
 out
 The output argument (DenseOneIndex with proper size).
 factor
 A scalar factor

new
()¶ Return a new oneindex object with the same nbasis

norm
()¶ Calculate L2 norm

permute_basis
(permutation)¶ Reorder the coefficients for a given permutation of basis functions.
Arguments:
 permutation
 An integer numpy array that defines the new order of the basis functions.

randomize
()¶ Fill with random normal data

set_element
(i, value)¶ Set a matrix element

sort_indices
(reverse=False)¶ Return indices of sorted arguments in decreasing order
Optional arguements
 reverse
 If True search order is reversed to increasing arguements

to_hdf5
(grp)¶ Dump this object in an h5py.Group
Arguments:
 grp
 An h5py.Group object.

trace
(begin0=0, end0=None)¶ Calculate trace
Optional arguments:
 begin0, end0
 Can be used to select a subblock of the object. When not given, the full range is used.

nbasis
¶ The number of basis functions

ndim
¶ The number of axes in the Nindex object.

shape
¶ The shape of the object

class
horton.matrix.dense.
DenseExpansion
(nbasis, nfn=None)¶ Bases:
horton.matrix.base.Expansion
Arguments:
 nbasis
 The number of basis functions.
Optional arguments:
 nfn
 The number of functions to store. Defaults to nbasis.
 do_energies
 Also allocate an array to store an energy corresponding to each function.

assign
(other)¶ Assign with the contents of another object
Arguments:
 other
 Another DenseExpansion or DenseTwoIndex object. If DenseTwoIndex, energies and occupations are set to zero.

assign_dot
(left, right)¶ Dot product of orbitals in a DenseExpansion and TwoIndex object
Arguments:
 left, right:
 An expansion and a twoindex object, or a twoindex and an expansion object.
The transformed orbitals are stored in self.

assign_occupations
(occupation)¶ Assign orbital occupations
Arguments:
 occupation
 The orbital occupations to be updated. An OneIndex instance

change_basis_signs
(signs)¶ Correct for different sign conventions of the basis functions.
Arguments:
 signs
 A numpy array with sign changes indicated by +1 and 1.

check_normalization
(overlap, eps=0.0001)¶ Check that the occupied orbitals are normalized.
Arguments:
 overlap
 The overlap twoindex operator
Optional arguments:
 eps
 The allowed deviation from unity, very loose by default.

check_orthonormality
(overlap, eps=0.0001)¶ Check that the occupied orbitals are orthogonal and normalized.
Arguments:
 overlap
 The overlap twoindex operator
Optional arguments:
 eps
 The allowed deviation from unity, very loose by default.

clear
()¶ Reset all elements to zero.

copy
()¶ Return a copy of the object

derive_naturals
(dm, overlap)¶ Arguments:
 dm
 A DenseTwoIndex object with the density matrix
 overlap
 A DenseTwoIndex object with the overlap matrix
Optional arguments:

error_eigen
(fock, overlap)¶ Compute the error of the orbitals with respect to the eigenproblem
Arguments:
 fock
 A DenseTwoIndex Hamiltonian (or Fock) operator.
 overlap
 A DenseTwoIndex overlap operator.
Returns: the RMSD error on the orbital energies

from_fock
(fock, overlap)¶ Diagonalize a Fock matrix to obtain orbitals and energies
This method updated the attributes
coeffs
andenergies
inplace.Arguments:
 fock
 The fock matrix, an instance of DenseTwoIndex.
 overlap
 The overlap matrix, an instance of DenseTwoIndex.

from_fock_and_dm
(fock, dm, overlap, epstol=1e08)¶ Combined Diagonalization of a Fock and a density matrix
This routine first diagonalizes the Fock matrix to obtain orbitals and orbital energies. Then, using first order (degenerate) perturbation theory, the occupation numbers are computed and, if needed, the the degeneracies of the Fock orbitals are lifted. It is assumed that the Fock and the density matrices commute. This method updated the attributes
coeffs
,energies
andoccupations
inplace.Arguments:
 fock
 The fock matrix, an instance of DenseTwoIndex.
 dm
 The density matrix, an instance of DenseTwoIndex.
 overlap
 The overlap matrix, an instance of DenseTwoIndex.
Optional arguments:
 epstol
 The threshold for recognizing degenerate energy levels. When two
subsequent energy levels are separated by an orbital energy less
than
epstol
, they are considered to be degenerate. When a series of energy levels have an orbital energy spacing between subsequent levels that is smaller thanepstol
, they are all considered to be part of the same degenerate group. For every degenerate set of orbitals, the density matrix is used to (try to) lift the degeneracy.

classmethod
from_hdf5
(grp)¶ Construct an instance from data previously stored in an h5py.Group.
Arguments:
 grp
 An h5py.Group object.

get_homo_energy
(offset=0)¶ Return a homo energy
Optional arguments:
 offset
 By default, the (highest) homo energy is returned. When this index is above zero, the corresponding lower homo energy is returned.

get_homo_index
(offset=0)¶ Return the index of a HOMO orbital.
Optional arguments:
 offset
 By default, the (highest) homo energy is returned. When this index is above zero, the corresponding lower homo energy is returned.

get_lumo_energy
(offset=0)¶ Return a lumo energy
Optional arguments:
 offset
 By default, the (lowest) lumo energy is returned. When this index is above zero, the corresponding higher homo energy is returned.

get_lumo_index
(offset=0)¶ Return the index of a LUMO orbital.
Optional arguments:
 offset
 By default, the (lowest) lumo energy is returned. When this index is above zero, the corresponding higher homo energy is returned.

imul
(other)¶ Inplace multiplication with other DenseOneIndex.
The attributes
energies
andoccupations
are not altered.Arguments:
 other
 A DenseOneIndex object.

itranspose
()¶ Inplace transpose

new
()¶ Return a new expansion object with the same nbasis and nfn

permute_basis
(permutation)¶ Reorder the coefficients for a given permutation of basis functions (rows).
Arguments:
 permutation
 An integer numpy array that defines the new order of the basis functions.

permute_orbitals
(permutation)¶ Reorder the coefficients for a given permutation of orbitals (columns).
Arguments:
 permutation
 An integer numpy array that defines the new order of the orbitals.

randomize
()¶ Fill with random normal data

rotate_2orbitals
(angle=0.7853981633974483, index0=None, index1=None)¶ Rotate two orbitals
Optional arguments:
 angle
 The rotation angle, defaults to 45 deg.
 index0, index1
 The orbitals to rotate, defaults to HOMO and LUMO,
The attributes
energies
andoccupations
are not altered.

rotate_random
()¶ Apply random unitary transformation distributed with Haar measure
The attributes
energies
andoccupations
are not altered.

swap_orbitals
(swaps)¶ Change the order of the orbitals using pairexchange
Arguments:
 swaps
 An integer numpy array with two columns where every row corresponds to one swap.
The attributes
energies
andoccupations
are also reordered.

to_dm
(out=None, factor=1.0, clear=True, other=None)¶ Compute the density matrix
Optional arguments:
 out
 An output density matrix (DenseTwoIndex instance).
 factor
 The density matrix is multiplied by the given scalar.
 clear
 When set to False, the output density matrix is not zeroed first.
 other
 Another DenseExpansion object to construct a transferdensity matrix.

to_hdf5
(grp)¶ Dump this object in an h5py.Group
Arguments:
 grp
 An h5py.Group object.

coeffs
¶ The matrix with the expansion coefficients

energies
¶ The orbital energies

homo_energy
¶ Return a homo energy
Optional arguments:
 offset
 By default, the (highest) homo energy is returned. When this index is above zero, the corresponding lower homo energy is returned.

lumo_energy
¶ Return a lumo energy
Optional arguments:
 offset
 By default, the (lowest) lumo energy is returned. When this index is above zero, the corresponding higher homo energy is returned.

nbasis
¶ The number of basis functions

nfn
¶ The number of orbitals (or functions in general)

occupations
¶ The orbital occupations

class
horton.matrix.dense.
DenseTwoIndex
(nbasis, nbasis1=None)¶ Bases:
horton.matrix.base.TwoIndex
Arguments:
 nbasis
 The number of basis functions. (Number of rows. Also number of columns, unless nbasis1 is given.)
Optional arguments:
 nbasis1
 When given, this is the number of columns (second index).
Note that by default the twoindex object is assumed to be Hermitian. Only when nbasis1 is given, this assumption is dropped.

assign
(other, ind=None)¶ Assign a new contents to the twoindex object
Arguments:
 other
 The new data, may be DenseTwoIndex, DenseOneIndex, a scalar value, or an ndarray.
 ind
 If provided, only these elements (of DenseOneIndex) are assigned

assign_diagonal
(value)¶ Set diagonal elements to value
Arguments:
 value
 Either a scalar or a DenseOneIndex object

assign_dot
(other, tf2)¶ Dot product of orbitals in a DenseExpansion and TwoIndex object
Arguments:
 other
 An expansion object with input orbitals
 tf2
 A twoindex object
The transformed array is stored in self.

assign_two_index_transform
(ao_integrals, exp0, exp1=None)¶ Perform two index transformation:
exp0.T * ao_integrals * exp0
Arguments:
 ao_integrals
 Something computed in the atomic orbital basis
 exp0
 The molecular orbitals.
Optional arguments:
 exp1
 When given, the following is computed:
exp0.T * ao_integrals * exp1

change_basis_signs
(signs)¶ Correct for different sign conventions of the basis functions.
Arguments:
 signs
 A numpy array with sign changes indicated by +1 and 1.

clear
()¶ Reset all elements to zero.

contract_to_one
(subscripts, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None)¶ Contract self to OneIndex.
Arguments:
 subscripts
ab>b
: contract first index,ab>a
: contract second index.
Optional arguments
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1
 Can be used to contract only a part of the twoindex object
Returns: the contracted oneindex object.

contract_two
(subscripts, two, begin0=0, end0=None, begin1=0, end1=None)¶ Compute the trace using with other twoindex objects
Arguments:
 subscripts
ab,ba
: trace of matrix product.ab,ab
: trace after elementwise multiplication (expectation_value). two
 A DenseTwoIndex instance
 begin0, end0, begin1, end1
 Can be used to select a subblock of the (other) object. When not given, the full range is used.

contract_two_to_one
(subscripts, other, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None)¶ Contract two TwoIndex objects to a one OneIndex.
Arguments:
 subscripts:
ab,ab>b
: contract first index.ab,ab>a
: contract second index. other
 The second DenseTwoIndex object.
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1
 Can be used to contract only a part of the other twoindex object. When not given, the full range is taken.
Returns: the contracted oneindex object.

contract_two_to_two
(subscripts, other, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None)¶ Contract two TwoIndex objects to a one twoIndex.
Arguments:
 subscripts:
ab,cb>ac
,ab,ca>cb
,ab,bc>ac
,ab,ac>cb
,ab,ac>bc
,ab,cb>ca
 other
 The second DenseTwoIndex object.
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1
 Can be used to select a subblock of the (other) object. When not given, the full range is used.
Returns: the contracted twoindex object.

copy
(begin0=0, end0=None, begin1=0, end1=None)¶ Return a copy of (a part of) the object
Optional arguments:
 begin0, end0, begin1, end1
 Can be used to select a subblock of the object. When not given, the full range is used. Only when the ranges are equal for both axes, an Hermitian twoindex may be returned.

copy_diagonal
(out=None, begin=0, end=None)¶ Copy (part of) the diagonal of the twoindex object
Optional arguments:
 out
 The output argument (DenseOneIndex with proper size).
 begin, end
 Can be used to select a range of the diagonal. If not given, then the entire diagonal is copied.

copy_slice
(ind, out=None)¶ Copy slice of the twoindex object into oneindex object
Arguments:
 ind
 2Tuple of 1dim arrays with row and column indices of TwoIndex object to be copied.
Optional arguments:
 out
 The output argument (DenseOneIndex with proper size).

diagonalize
()¶ Return eigenvalues of twoindex object

distance_inf
(other)¶ The infinity norm distance between self and other
Arguments:
 other
 A DenseTwoIndex instance.

classmethod
from_hdf5
(grp)¶ Construct an instance from data previously stored in an h5py.Group.
Arguments:
 grp
 An h5py.Group object.

get_element
(i, j)¶ Return a matrix element

iabs
()¶ Inplace absolute values

iadd
(other, factor=1.0, begin0=0, end0=None, begin1=0, end1=None, transpose=False)¶ Add another DenseTwoIndex object inplace, multiplied by factor. If begin0, end0, begin1, end1 are specified, other is added to the selected range.
Arguments:
 other
 A DenseTwoIndex, DenseOneIndex instance or float to be added
Optional arguments:
 factor
 A scalar factor
 begin0, end0, begin1, end1
 When given, specify the ranges where the contribution will be added. When not given, the full range is used.

iadd_contract_two_one
(subscripts, two, one, factor=1.0, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None)¶ Inplace addition of a contraction of a two and oneindex object
Arguments:
 two, one
 The two and oneindex objects, respectively. Instances of
 subscripts
ab,b>ab
: contract with the first index of the twoindex object.ab,a>ab
: contract with the second index of the twoindex.ba,a>ab
: contract with the second index and transpose. object. factor
 The term added is scaled by this factor.
 begin0, end0, begin1, end1
 Can be used to select a subblock of the twoindex object (two). When not given, the full range is used.
 begin2, end2
 Can be used to select a subblock of the oneindex object (one). When not given, the full range is used.

iadd_dot
(other0, other1, factor=1.0, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Inplace addition of dot product:
other0 * other1
Arguments:
 other0, other1
 Twoindex objects that go into the kronecker product.
Optional arguments:
 factor
 The term added is scaled by this factor.
 begin0, end0, begin1, end1
 Can be used to select a subblock of the other0 object. When not given, the full range is used.
 begin2, end2, begin3, end3
 Can be used to select a subblock of the other1 object. When not given, the full range is used.

iadd_dott
(other0, other1, factor=1.0)¶ Inplace addition of dot product:
other0 * other1.T
Arguments:
 other0, other1
 Twoindex objects that go into the kronecker product.
Optional arguments:
 factor
 The term added is scaled by this factor.

iadd_kron
(other0, other1, factor=1.0)¶ Inplace addition of kronecker product of two other DenseTwoIndex
Arguments:
 other0, other1
 Twoindex objects that go into the kronecker product.
Optional arguments:
 factor
 The term added is scaled by this factor.

iadd_mult
(other0, other1, factor=1.0, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None, transpose0=False, transpose1=False)¶ Inplace addition of multiplication:
other0 * other1
Arguments:
 other0, other1
 Twoindex objects that go into the product.
Optional arguments:
 factor
 The term added is scaled by this factor.
 begin0, end0, begin1, end1
 Can be used to select a subblock of the other1 object. When not given, the full range is used.
 begin2, end2, begin3, end3
 Can be used to select a subblock of the other0 object. When not given, the full range is used.
 transpose0, transpose1
 Can be used to select transpose of other0, other1 objects

iadd_one_mult
(other0, other1, factor=1.0, transpose0=False, transpose1=False)¶ Inplace addition of multiplication:
other0 * other1
Arguments:
 other0, other1
 Oneindex objects that go into the product.
Optional arguments:
 factor
 The term added is scaled by this factor.
 transpose0, transpose1
 Can be used to select transpose of oneindex objects.

iadd_outer
(other0, other1, factor=1.0)¶ Inplace addition of outer product of two other DenseTwoIndex
Arguments:
 other0, other1
 Twoindex objects that go into the outer product. They are raveled before taking the outer product.
Optional arguments:
 factor
 The term added is scaled by this factor.

iadd_shift
(lshift)¶ Add positive shift to elements. If negative subtract shift
Arguments:
 lshift
 A scalar used to augment the matrix elements.

iadd_slice
(other, factor=1.0, begin0=0, end0=None, begin1=0, end1=None)¶ Add slice of another DenseTwoIndex object inplace, multiplied by factor. The slice of other is added to the full range of the array.
Arguments:
 other
 A DenseTwoIndex instance to be added
Optional arguments:
 factor
 A scalar factor
 begin0, end0, begin1, end1
 When given, specify the ranges of contribution to be added. When not given, the full range is used.

iadd_t
(other, factor=1.0, begin0=0, end0=None, begin1=0, end1=None)¶ See
DenseTwoIndex.iadd()
, transpose=True

iadd_tdot
(other0, other1, factor=1.0)¶ Inplace addition of dot product:
other0.T * other1
Arguments:
 other0, other1
 Twoindex objects that go into the kronecker product.
Optional arguments:
 factor
 The term added is scaled by this factor.

idot
(other)¶ Inplace dot product: self = self * other
Arguments:
 other
 The other array

imul
(other, factor=1.0, begin0=0, end0=None, begin1=0, end1=None)¶ Inplace elementwise multiplication:
self *= other * factor
Arguments:
 other
 A DenseTwoIndex instance.
Optional arguments:
 factor
 The twoindex object is scaled by this factor.
 begin0, end0, begin1, end1
 Can be used to select a subblock of the (other) object. When not given, the full range is used.

imul_t
(other, factor=1.0, begin0=0, end0=None, begin1=0, end1=None)¶ Inplace elementwise multiplication:
self *= other.T * factor
Arguments:
 other
 A DenseTwoIndex instance.
Optional arguments:
 factor
 The twoindex object is scaled by this factor.
 begin0, end0, begin1, end1
 Can be used to select a subblock of the (other) object. When not given, the full range is used.

inner
(vec0, vec1)¶ Compute an inner product of two vectors using the twoindex as a metric
Arguments:
 vec0, vec1
 The vectors, either DenseOneIndex or numpy arrays.

inverse
()¶ Return the inverse of twoindex object

iortho
()¶ Inplace orthogonalization
Arguments:

is_shape_symmetric
(symmetry)¶ Check whether the symmetry argument matches the shape

is_symmetric
(symmetry=2, rtol=1e05, atol=1e08)¶ Check the symmetry of the array.
Optional arguments:
 symmetry
 The symmetry to check. See Handling of index symmetry for more details.
 rtol and atol
 relative and absolute tolerance. See to
np.allclose
.

iscale
(factor)¶ Inplace multiplication with a scalar
Arguments:
 factor
 A scalar factor.

itranspose
()¶ Inplace transpose

new
()¶ Return a new twoindex object with the same nbasis (and nbasis1)

permute_basis
(permutation)¶ Reorder the coefficients for a given permutation of basis functions.
The same permutation is applied to all indexes.
Arguments:
 permutation
 An integer numpy array that defines the new order of the basis functions.

randomize
()¶ Fill with random normal data

set_element
(i, j, value, symmetry=2)¶ Set a matrix element
Arguments:
 i, j
 The matrix indexes to be set
 value
 The value to be assigned to the matrix element.
Optional arguments:
 symmetry
 When 2 (the default), the element (j,i) is set to the same value. When set to 1 the opposite offdiagonal is not set. See Handling of index symmetry for more details.

sqrt
()¶ Return the real part of the square root of twoindex object

sum
(begin0=0, end0=None, begin1=0, end1=None)¶ Return the sum of all elements (in the selected range)
Optional arguments:
 begin0, end0, begin1, end1
 Can be used to select a subblock of the object to be contracted.

symmetrize
(symmetry=2)¶ Symmetrize inplace
Optional arguments:
 symmetry
 The symmetry to impose. See Handling of index symmetry for more details.

to_hdf5
(grp)¶ Dump this object in an h5py.Group
Arguments:
 grp
 An h5py.Group object.

trace
(begin0=0, end0=None, begin1=0, end1=None)¶ Return the trace of the twoindex object.
Optional arguments:
 begin0, end0, begin1, end1
 Can be used to select a subblock of the object to be contracted.

nbasis
¶ The number of basis functions

nbasis1
¶ The other size of the twoindex object

ndim
¶ The number of axes in the Nindex object.

shape
¶ The shape of the object

class
horton.matrix.dense.
DenseThreeIndex
(nbasis, nbasis1=None, nbasis2=None)¶ Bases:
horton.matrix.base.ThreeIndex
Arguments:
 nbasis
 The number of basis functions.

assign
(other)¶ Assign with the contents of another object
Arguments:
 other
 Another DenseThreeIndex object or a scalar.

change_basis_signs
(signs)¶ Correct for different sign conventions of the basis functions.
Arguments:
 signs
 A numpy array with sign changes indicated by +1 and 1.

clear
()¶ Reset all elements to zero.

contract_to_two
(subscripts, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None)¶ Contract self to TwoIndex.
Arguments:
 subscripts
abc>ac
: contract first index.
Optional arguments
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2
 Can be used to contract only a part of the threeindex object. When not given, the full range is used.
Returns: the contracted twoindex object.

contract_two_to_three
(subscripts, two, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None)¶ Contracts with twoindex to obtain threeindex.
Arguments:
 subscripts
 One of
abc,ad>cdb
,abc,ad>cbd
,abc,da>dbc
,abc,da>bdc
 inp
 A DenseTwoIndex input object.
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2
 Can be used to contract only a part of the threeindex object

contract_two_to_two
(subscripts, inp, out=None, factor=1.0, clear=True)¶ Contracts self with twoindex to obtain twoindex.
Arguments:
 subscripts
 One of
abc,ab>ac
,abc,ab>ca
,abc,bc>ba
,abc,bc>ab
,abc,cb>ac
.  inp
 A DenseTwoIndex input object.
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()

copy
(begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None)¶ Return a copy of (a part of) the object
Optional arguments:
 begin0, end0, begin1, end1, begin2, end2
 Can be used to select a subblock of the object. When not given, the full range is used.

classmethod
from_hdf5
(grp)¶ Construct an instance from data previously stored in an h5py.Group.
Arguments:
 grp
 An h5py.Group object.

get_element
(i, j, k)¶ Return a matrix element

iadd
(other, factor=1.0)¶ Add another DenseThreeIndex object inplace, multiplied by factor
Arguments:
 other
 A DenseThreeIndex instance to be added
Optional arguments:
 factor
 The added term is scaled by this factor.

iadd_expand_two_one
(subscripts, two, one, factor=1.0)¶ Inplace addition of expanded twoindex and oneindex to threeindex.
Arguments:
 subscripts
 Contraction type:
ab,c>cab
orab,c>acb
.  two
 A DenseTwoIndex object.
 one
 A DenseOneIndex object.
Optional arguments:
 factor
 The added term is scaled by this factor.

iadd_expand_two_two
(subscripts, two0, two1, factor=1.0, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Inplace addition of expanded twoindex and twoindex to threeindex.
Arguments:
 subscripts
 Contraction type:
ac,bc>abc
,ab,bc>abc
,ab,ac>acb
,cb,ac>acb
,ac,ab>abc
,ab,ac>abc
 two0, two1
 A DenseTwoIndex object.
Optional arguments:
 factor
 The added term is scaled by this factor.
 begin0, end0, begin1, end1
 Can be used to contract only a part of the two1 object. When not given, the full range is used.
 begin2, end2, begin3, end3
 Can be used to contract only a part of the two0 object. When not given, the full range is used.

iadd_slice
(other, factor=1.0, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None)¶ Add slice of another DenseThreeIndex object inplace, multiplied by factor
Arguments:
 other
 A DenseThreeIndex instance to be added
Optional arguments:
 factor
 The added term is scaled by this factor.
 begin0, end0, begin1, end1, begin2, end2
 Can be used to add only a part of the threeindex object

iscale
(factor)¶ Inplace multiplication with a scalar
Arguments:
 factor
 A scalar factor.

new
()¶ Return a new threeindex object with the same nbasis

permute_basis
(permutation)¶ Reorder the coefficients for a given permutation of basis functions.
Arguments:
 permutation
 An integer numpy array that defines the new order of the basis functions.

randomize
()¶ Fill with random normal data

set_element
(i, j, k, value)¶ Set a matrix element

to_hdf5
(grp)¶ Dump this object in an h5py.Group
Arguments:
 grp
 An h5py.Group object.

nbasis
¶ The number of basis functions

nbasis1
¶ The number of basis functions

nbasis2
¶ The number of basis functions

ndim
¶ The number of axes in the Nindex object.

shape
¶ The shape of the object

class
horton.matrix.dense.
DenseFourIndex
(nbasis, nbasis1=None, nbasis2=None, nbasis3=None)¶ Bases:
horton.matrix.base.FourIndex
Arguments:
 nbasis
 The number of basis functions.
Optional arguments:
nbasis1, nbasis2, nbasis3

assign
(other)¶ Assign with the contents of another object
Arguments:
 other
 Another DenseFourIndex object or a np ndarrray.

assign_four_index_transform
(ao_integrals, exp0, exp1=None, exp2=None, exp3=None, method='tensordot')¶ Perform four index transformation.
Arguments:
 oa_integrals
 A DenseFourIndex with integrals in atomic orbitals.
 exp0
 A DenseExpansion object with molecular orbitals
Optional arguments:
 exp1, exp2, exp3
 Can be provided to transform each index differently.
 method
 Either
einsum
ortensordot
(default).

change_basis_signs
(signs)¶ Correct for different sign conventions of the basis functions.
Arguments:
 signs
 A numpy array with sign changes indicated by +1 and 1.

clear
()¶ Reset all elements to zero.

contract_four
(subscripts, other, factor=1.0, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Compute the trace with other fourindex object
Arguments:
 subscripts
abcd,abcd
,abcd,adcb
,abcd,acbd
,abcd,acdb
 other
 A DenseFourIndex instance
Optional arguments:
 factor
 A scalar factor
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the other object. When not given, the full range is used.

contract_four_to_four
(subscripts, four, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Contracts with a fourindex object to obtain a fourindex object.
Arguments:
 subscripts
 Any of
abcd,cedf>abfe
,abcd,cefd>abfe
,abcd,cedf>feab
,abcd,cefd>feab
,abcd,eadf>fbce
,abcd,eadf>cefb
,abcd,aedf>cbfe
,abcd,aedf>fecb
,abcd,acef>ebfd
,abcd,bdef>aecf
,abcd,cedf>abef
,abcd,cefd>abef
,abcd,cedf>efab
,abcd,cefd>efab
,abcd,eadf>ebcf
,abcd,eadf>cfeb
,abcd,eadf>cbef
,abcd,eadf>efcb
,abcd,eafd>cbef
,abcd,eafd>efcb
 four
 An instance of DenseFourIndex.
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the other fourindex object (four). When not given, the full range is used.

contract_four_to_two
(subscripts, four, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Contracts with a fourindex object to obtain a twoindex object.
Arguments:
 subscripts
 Any of
abcd,aced>be
,abcd,acde>be
,abcd,aced>eb
,abcd,acde>eb
,abcd,ecbd>ae
,abcd,ecdb>ae
,abcd,ecdb>ea
,abcd,ecbd>ea
,abcd,acbe>ed
,abcd,aceb>ed
,abcd,aebd>ce
,abcd,aedb>ce
 four
 An instance of DenseFourIndex.
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the other fourindex object (four). When not given, the full range is used.

contract_three_to_three
(subscripts, three, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None)¶ Contracts with a threeindex object to obtain a threeindex object.
Arguments:
 subscripts
 Any of
abcd,ace>ebd
,abcd,ebd>ace
 three
 An instance of DenseThreeIndex.
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2
 Can be used to select a subblock of the threeindex object (three). When not given, the full range is used.

contract_to_two
(subscripts, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Contract fourindex object to twoindex object
Arguments:
 subscripts
abcb>ac
,abbc>ac
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the object. When not given, the full range is used.

contract_two
(subscripts, other, factor=1.0, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Compute diagonal trace with twoindex objects
Arguments:
 subscripts
aabb,ab
. other
 A DenseTwoIndex instance
Optional arguments:
 factor
 A scalar factor. See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the fourindex object. When not given, the full range is used.

contract_two_to_four
(subscripts, two, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None, begin4=0, end4=None, begin5=0, end5=None)¶ Contracts with a twoindex object to obtain a fourindex object.
Arguments:
 subscripts
 Any of
abcd,cd>acbd
,abcd,cd>acdb
,abcd,cb>acdb
,abcd,cb>acbd
,abcd,ab>acbd
,abcd,ab>acdb
,abcd,ad>acbd
,abcd,ad>acdb
,abcd,ad>abcd
,abcd,ad>abdc
,abcd,bd>abcd
,abcd,bd>abdc
,abcd,bc>abdc
,abcd,bc>abcd
,abcd,ac>abcd
,abcd,ac>abdc
,abcd,bd>cabd
,abcd,bc>dabc
,abcd,ca>cabd
,abcd,da>dabc
,abcd,dc>dabc
,abcd,ba>dabc
,aabc,dc>adbc
,aabc,db>adbc
,abcc,ad>abcd
,abcc,bd>abcd
,abcd,bc>acbd
,abcd,eb>aecd
,abcd,eb>cdae
,abcd,ed>ceab
,abcd,ed>abce
,abcd,ae>cdeb
,abcd,ae>ebcd
,abcd,ce>edab
,abcd,ce>abed
,abcd,ab>abcd
,abcd,cb>abcd
,abcd,ec>eadb
,abcd,ec>dbea
,abcd,ae>cedb
,abcd,ae>dbce
,abcd,ec>ebad
,abcd,ed>ebac
,abcd,ec>adeb
,abcd,ed>aceb
,abcd,ae>debc
,abcd,ae>cebd
,abcd,ae>bcde
,abcd,ae>bdce
 two
 An instance of DenseTwoIndex.
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the fourindex object. When not given, the full range is used.
 begin4, end4, begin5, end5
 Can be used to select a subblock of the twoindex object (two). When not given, the full range is used.

contract_two_to_three
(subscripts, two, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Contracts with a twoindex object to obtain a threeindex object.
Arguments:
 subscripts
 Any of
aabc,ad>bcd
,'abcd,ac>bdc
,abcd,ad>bcd
,abcd,bc>abd
,abcd,ac>abd
,abcc,dc>dab
,abcd,ac>bcd
,abcc,cd>dab
,abcc,dc>abd
,abcb,db>adc
,abcc,dc>adb
,abcb,db>dac
,abcc,cd>abd
 two
 An instance of DenseTwoIndex.
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the fourindex object. When not given, the full range is used.

contract_two_to_two
(subscripts, two, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None, begin4=0, end4=None, begin5=0, end5=None)¶ Contract self with a twoindex to obtain a twoindex.
Arguments:
 subscripts
 Any of
abcd,bd>ac
(direct),abcd,cb>ad
(exchange),aabb,cb>ac
,abcc,bc>ab
,aabc,ab>bc
,aabc,ac>bc
,abcc,ac>ab
,abcb,cb>ac
,abcb,ab>ac
,abcc,bc>ba
,abcc,bc>ab
,abcd,ac>db
,abcd,ad>cb
,abcd,ac>bd
,abcd,ad>bc
,abcd,ab>cd
 two
 The input twoindex object. (DenseTwoIndex)
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the fourindex object. When not given, the full range is used.
 begin4, end4, begin5, end5
 Can be used to select a subblock of the twoindex object (two). When not given, the full range is used.

copy
(begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Return a copy of (a part of) the object
Optional arguments:
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the object. When not given, the full range is used.

classmethod
from_hdf5
(grp)¶ Construct an instance from data previously stored in an h5py.Group.
Arguments:
 grp
 An h5py.Group object.

get_element
(i, j, k, l)¶ Return a matrix element

iadd
(other, factor=1.0)¶ Add another DenseFourIndex object inplace, multiplied by factor
Arguments:
 other
 A DenseFourIndex instance to be added
Optional arguments:
 factor
 The added term is scaled by this factor.

iadd_exchange
()¶ Inplace addition of its own exchange contribution

iadd_expand_three_to_four
(axis, three, factor=1.0)¶ Expand threeindex object along one axis and add it to fourindex object.
Arguments:
 axis
 Any of
1312
(abac += abc),1212
(abca += abc),0202
(abcb += abc),0302
(abbc += abc),0201
(abcb += acb). The expansion is performed for repeated indices in the fourindex object  three
 An instance of DenseThreeIndex.
Optional arguments:
 factor
 See
DenseLinalgFactory.einsum()

iadd_expand_two_to_four
(axis, two, factor=1.0, begin0=0, end0=None, begin1=0, end1=None)¶ Expand twoindex object along two axes and add it to fourindex object.
Arguments:
 axis
 Any of
13
(abac += bc),02
(abcb += ac),03
(abbc += ac),12
(abca += bc),diag
(abab += ab). The expansion is performed for the repeated indices in the fourindex object  two
 An instance of DenseTwoIndex.
Optional arguments:
 factor
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1
 Can be used to select a subblock of the twoindex object (two). When not given, the full range is used.

imul
(other, factor=1.0)¶ Inplace elementwise multiplication:
self *= other * factor
Arguments:
 other
 A DenseFourIndex instance.
Optional arguments:
 factor
 The fourindex object is scaled by this factor.

is_shape_symmetric
(symmetry)¶ Check whether the symmetry argument matches the shape

is_symmetric
(symmetry=8, rtol=1e05, atol=1e08)¶ Check the symmetry of the array.
Optional arguments:
 symmetry
 The symmetry to check. See Handling of index symmetry for more details. In addition to 1, 2, 4, 8, also ‘cdab’ is supported.
 rtol and atol
 relative and absolute tolerance. See to
np.allclose
.

iscale
(factor)¶ Inplace multiplication with a scalar
Arguments:
 factor
 A scalar factor.

itranspose
()¶ Inplace transpose:
0,1,2,3 > 1,0,3,2

new
()¶ Return a new fourindex object with the same nbasis

permute_basis
(permutation)¶ Reorder the coefficients for a given permutation of basis functions.
Arguments:
 permutation
 An integer numpy array that defines the new order of the basis functions.

randomize
()¶ Fill with random normal data

reshape
(shape)¶ Reshape array
Optional arguments:
 shape
 List containing the new dimension of each axis.

set_element
(i, j, k, l, value, symmetry=8)¶ Set a matrix element
Arguments:
 i, j, k, l
 The matrix indexes to be set
 value
 The value to be assigned to the matrix element.
Optional arguments:
 symmetry
 The level of symmetry to be enforced when setting the matrix element. See Handling of index symmetry for more details.

slice_to_four
(subscripts, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Returns a fourindex contraction of the fourindex object.
Arguments:
 superscripts
 Any of
abcd>abcd
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the object. When not given, the full range is used.

slice_to_three
(subscripts, out=None, factor=1.0, clear=True)¶ Returns a threeindex contraction of the fourindex object.
Arguments:
 superscripts
 Any of
abcc>bac
,abcc>abc
,abcb>abc
,abbc>abc
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()

slice_to_two
(subscripts, out=None, factor=1.0, clear=True, begin0=0, end0=None, begin1=0, end1=None, begin2=0, end2=None, begin3=0, end3=None)¶ Returns a twoindex contraction of the fourindex object.
Arguments:
 superscripts
 Any of
aabb>ab
,abab>ab
,abba>ab
Optional arguments:
 out, factor, clear
 See
DenseLinalgFactory.einsum()
 begin0, end0, begin1, end1, begin2, end2, begin3, end3
 Can be used to select a subblock of the object. When not given, the full range is used.

sum
()¶ Return the sum of all elements

symmetrize
(symmetry=8)¶ Symmetrize inplace
Optional arguments:
 symmetry
 The symmetry to impose. See Handling of index symmetry for more details.

to_hdf5
(grp)¶ Dump this object in an h5py.Group
Arguments:
 grp
 An h5py.Group object.

nbasis
¶ The number of basis functions

nbasis1
¶ The number of basis functions

nbasis2
¶ The number of basis functions

nbasis3
¶ The number of basis functions

ndim
¶ The number of axes in the Nindex object.

shape
¶ The shape of the object