Input Documentation

General Structure

• Special characters
• Using the C preprocessor (CPP) with MCTDH
• Keyword placement
• Arguments to keywords
• Units

Special characters

Using the C preprocessor (CPP) with MCTDH

As an example, take a look at the input files allene_e.inp and allene_b.inp with common settings in the include files allene_common1.inc and allene_common2.inc.
These examples are also listed in the examples section below.

Keyword placement

The order and placement of the keywords within a section is usually arbitrary. Exceptions are described below.

Arguments to keywords.

    rd    sin     36   3.800    5.60

CAP_rd = CAP [ 5.0  0.357    3 ]

Units.

SECTIONS

The input sections reflect the structure of the program. The RUN-SECTION defines what sort of calculation is desired. Depending on what is requested here the remaining sections provide the required information. The various sections are listed below. There is no pre-defined order for the sections.
 

Below are tables of keywords for each input section. The number and type of arguments is specified. The type is S for character string, R for real number, and I for integer. e.g.
keyword = S, R
indicates that the keyword takes two arguments, the first is a string, the second a real number. Default values for keywords and arguments are also listed.

RUN-SECTION

There are 4 levels of calculation types, reflecting the four stages of a calculation. Only one run-type keyword of a particular level is allowed. All necessary lower level keywords are automatically included. Thus
propagation
or
gendvr genoper geninwf propagation
are equivalent.

Notes on Improved Relaxation.
When the keyword relaxation is given, a normal relaxation, i.e. a propagation in negative imaginary time, is performed. However, relaxation=<...> will enforce an improved relaxation run, where the A-vector is not determined by relaxation but by diagonalisation of the Hamiltonian in the basis of the SPFs. Improved relaxation requires that one uses a CMF integrator scheme in the fix or varphi mode. (NB: The keyword CMF is interpreted as CMF/varphi when the runtype is improved relaxation. Otherwise CMF is always interpreted as CMF/var, see INTEGRATOR-SECTION. It is recommended to use CMF without any extension.
For diagonalising the Hamiltonian in the basis of the SPFs one may use the SIL "integrator" (now, of coarse, a diagonaliser), or, which is usually the better choice, a Davidson routine. There are actually several Davidson routines implemented, called DAV, rDAV, rrDAV, and cDAV. See INTEGRATOR-SECTION for details. In short, DAV is for (general) hermitian Hamiltonians, rDAV and rrDAV are for real-symmetric Hamiltonians, and cDAV is for complex non-hermitian Hamiltonians, i.e. for computing resonances. Actually, the rrDAV keyword calls the rDAV routine but real arithmetic is then used for the H*A operation. rDAV alone stores the A-vectors as real. All DAV "integrators", except cDAV, can also be used in block form, i.e. for converging a set of eigenstates simultaneously. If the keyword relaxation=0 is given, the algorithm will simply converge to the ground state, or, in a block improved relaxation run, to a set of states of lowest energies. If relaxation=I is used, with 0 < I < 900, then the I-th state (counting from 0) of the very first diagonalisation is taken, and for the following diagonalisations the state selection is done with follow (see below). However, the I-th state of the Davidson (or Lanczos) matrix will in general not be the I-th state of the Hamiltonian, but a higher one. To minimize this effect there is the keyword follow which forces the routine to take as many Davidson/Lanczos iterations as allowed by input. In any case, relaxation=0 with I > 0 is only for testing, to compute excited states one should use relaxation=follow or relaxation=lock. With follow the Davidson diagonalisation takes that eigenvector which is closest to its start vector, i.e. to the A-vector of the initial WF in case of the very first diagonalisation, or the selected eigenvector of the previous diagonalisation. With lock the Davidson diagonalisation takes that eigenvector which is closest to the initial WF as defined in the Init_WF-Section. I.e. it accounts for changes in the SPFs. lock is hence the safer choice for finding excited states, but it is numerically more demanding than follow. lock requires that the keyword cross is given in the Run-Section.
The rDAV routine allows to additionally use the keywords skip1dav, quad, olsen, backrotate, and cn, (where n denotes an integer, e.g. c6) as arguments to relaxation .
The keyword skip1dav lets the program skip the first Davidson diagonalisation, i.e. the calculation starts with orbital relaxation. This is only useful, if a relaxation is re-started and the WF is read in via the file statement. Then the A-vector was already obtained by diagonalisation and it does not make sense to diagonalise again. However, when build or autoblock is used, skip1dav should not be set!
The keyword quad lets the program switch to use a quadratic variational principle (i.e. (H-E)**2 is diagonalised within the space of Davidson-vectors rather than H).
With the keyword olsen the program applies the Olsen correction to the residual Davidson vector. This is useful particularly when a preconditioner is used. (Keyword precon). The Olsen correction in essence turns the Davidson algorithm into a Jacobi-Davidson one.
The keyword backrotate lets the program rotate the SPFs after relaxation, such that they have maximal overlap with the orbitals prior to the orbital relaxation step. This sometimes improves the overlap with the previous A-vector.
The keyword cn lets the program perform n orbital relaxation steps, before a new diagonalisation step is done.

When a calculation starts converging to a desired state but after a few iterations jumps to another state, then the numbers of single-particle functions are too small. It may also be necessary to increase the Davidson order. Inspect the rlx_info file (by running the script rdrlx) to see what Davidson orders are actually taken. Also inspect the update file. If a "*" appears at the beginning of a line, then the corresponding update time was too large.

The file rlx_info contains a lot of information on the relaxation process. Use the script rdrlx to read it. The first line of the output of the script rdrlx is labeled with a negative time. This line shows the expectation value of the Hamiltonian with respect to the provided initial state. The second line, labeled with time=0, shows the energy after the first diagonalisation, but without SPF relaxation. The following lines refer to SPF relaxation and subsequent diagonalisation.
With the aid of the keyword rlxunit one may set the energy-unit for the output to the rlx_info file. One also may apply an energy-shift (e.g. subtract the ground-state energy). See the table Keywords associated with a propagation or relaxation calculation below.

For an example input see the files co2_gs.inp and co2_asym.inp which generate the ground state and the asymmetric stretch excited state, respectively. See also the User's Guide, section 3.4 .

A block-improved-relaxation is performed when the packets keyword is given in the SPF-Basis-Section. The packets keyword must have the argument single-set, which ensures that the packages are treated in single-set; e.g.: packets=6,single-set. There still may be a multi-set keyword, if there are electronic states as well. The DAV, RDAV or RRDAV keyword must be given in the Integrator-Section and the keywords lock,quad,backrotate,cn are not implemented for block-improved-relaxation. For an example input on block-improved-relaxation see blkHONO.inp.

Notes on Diagonalisation.
With the keyword diagonalisation the program will perform a diagonalisation of the Hamiltonian using a straight-forward (non sophisticated) Lanczos algorithm. For this, the wave function is in exact format. Although the Lanczos algorithm is part of the MCTDH package, it is not based on the MCTDH method and hence does not inherit its efficiency. Moreover, diagonalisation is not parallelized. Diagonalisation has been implemented to check filter-diagonalisation. We do not recommend the use of diagonalisation, in particular for larger systems. It will not work for wave functions with more than 10⁹ grid points.

The exact keyword can be added to any run-type keyword. For example,
propagation exact
will result in a numerically exact wavepacket propagation (i.e. the wavefunction will be represented in the full product primitive basis). Similarly,
geninwf exact
will set up a wavepacket for a numerically exact calculation.

These keywords are equivalent to a run-type keyword of the level given.

The tfinal keyword must be given. All other keywords are optional. The following default values are used.

The following keywords define the data calculated and saved. If keywords are omitted, data will not be calculated.

OPERATOR-SECTION

If no OPERATOR-SECTION is included, the program looks for the operator information in the input file. This can be useful if a small model operator is studied. A full log of the operator is then automatically output.

ML-BASIS-SECTION

For a ML-MCTDH calculation the SPF-BASIS-SECTION must be replaced by a ML-BASIS-SECTION which defines the ML-tree. The top layer (up-most A-vector) is indicated by 0> and the following ones by 1>, 2>, etc. The numbers following these symbols gives the numbers of SPFs. E.g.

  0> 5 5 5  

   2> [q1]
 

   2> [q1,q2]
 

 mlbasis-section
 0> 5 5 5
    1> 5 5
        2> [q1]
        2> [q2]
    1> 5 5
        2> [q3]
        2> [q4]
    1> 5 5
        2> [q5]
        2> [q6]
 end-mlbasis-section
   

 ML-Basis-Section
 0> 2 2
   # Electronic
   1> [el]
   # Vibrations
   1> 4 4
     # Main system
     2> 4 4
       3> [v10a v6a]
       3> [v1 v9a v8a]
     # Bath
     2> 3 3
       3> 2 2 2
         4> [v2 v6b v8b]
         4> [v4 v5 v3]
         4> [v16a v12 v13]
       3> 3 2 2
         4> 2 2
           5> [v19b]
           5> [v18b]
         4> 2 2
           5> [v18a v14]
           5> [v19a v17a]
         4> 2 2
           5> [v20b v11 v7b]
           5> [v16b]
 end-mlbasis-section
   

SPF-BASIS-SECTION

The following lines define the single-particle function basis to be used in a wave function calculation. The input defines firstly how many degrees of freedom are contained within a mode. Secondly, the number of spfs are given; a list being needed for a multi-set basis. The format is:

mode_label1 , mode_label2 , ... = nspf1 , nspf2 , ...

For example for a 3-mode system, with labels X, Y and Z,

to define an spf basis of 3 functions per mode,

spf-basis-section
    X = 3
    Y = 3
    Z = 3
end-spf-basis-section

spf-basis-section
    X = 3    Y = 3    Z = 3
end-spf-basis

spf-basis
    X, Y = 3    Z = 3
end-spf-basis-section

spf-basis-section
multi-set
    X , Y = 3 , 2
    Z = 3,2
end-spf-basis-section

Note: The electronic SPF-Basis is not to be specified in the spf-basis-section, as it is always complete. The electronic mode will always be the last mode in a single-set run.

If many electronic states are present, the definition of single-particle functions for a mode can be continued on a second line by using a continuation mark %, e.g.

spf-basis-section
    X    =   5 , 5 , 5 , 5 , 6 , 10 , 5 , 2 , 2 , %
             5 , 5 , 2 , 2 , 3
end-spf-basis-section

If for symmetry reasons one set of SPFs is always identical to another one, the latter set need not to be propagated numerically. In such a situation, e.g. H₂O in valence coordinates and for a symmetric initial state, one may tell the program not to propagate the second identical set of SPFs. This is done by specifying with the id keyword the mode to which the present mode is identical. E.g.:

spf-basis-section
    R1    = 8
    R2    = id,1
    Theta = 9
end-spf-basis-section

The following keywords, if given, define multi-state or muti-packet runs. Note: A SPF-BASIS-SECTION does not need to exist for an exact calculation, except if packets is specified.

PRIMITIVE-BASIS-SECTION

mode_label   basis_type   basis_size    parameters

mode_label is an alphanumeric string (case sensitive) labelling the degree of freedom.

basis_type must be one of the following:
 

basis_size is an integer specifying the primitive basis size, e.g. grid points or vector elements etc. Note that for an FFT-representation basis_size (i.e. gdim in the program) must have a prime factor decomposition with only 2's, 3's and 5's but should have a decomposition with only 2's and 3's for optimal performance, i. e. basis_size = 2^m3ⁿ where m and n are positive integers. One can use the utility script find235.py to find numbers that have this required form.
Note also that for the exp-DVR the basis_size can be even or odd, but odd may be preferred as for an even number of grid points the covered momentum space is the same as when using one grid-point less.
For a sphFBR-representation, basis_size is not required in input. The number of basis functions is calculated by the program itself, according to the type of basis (see parameter description).
For a K-DVR basis_size is not required in input. It will be calculated from the kmin,kmax parameters given in this section. NB: The basis_size is called gdim.

The parameters to be input depend on the basis type as follows:

Remarks on harmonic oscillator DVR:

The HO-DVR depends only on the product hofreq*homass. If the homass entry is missing, the program sets homass to 1. Alternatively, one may specify the first and last grid-point. The program then computes the corresponding product hofreq*homass.
Example: The following lines are equivalent.

    Y    HO    32    0.00       0.10        1822.89
    Y    HO    32    0.00       2.721,eV    1822.89,au
    Y    HO    32    0.00       0.1,au      1.0,AMU
    Y    HO    32    0.00   21947.46,cm-1   1.0,AMU
    Y    HO    32    xi-xf     -0.528       0.528

Remarks on Laguerre DVR:

The Lagua-DVR is build from the basis functions:
phi(n,x) = Sqrt((n-1)!/(n+a-1)!) * x^(a/2) * exp(-x/2) * L(n-1,a,x) ,
where a = 1,2,3 or 4 (Lagu1 - Lagu4). Hence, the boundary condition for x -> 0 is phi(x) ~ x^(a/2). (With the aid of the parameters x0 and b the coordinates may be shifted and scaled, i.e. phi(x) <- phi((x-x0)/b)) ). The distribution of the grid points is very uneven, being very dense for small x and widely spaced for large x. The matrix of second derivatives may have very large negative eigenvalues. These will slow down the integrator. The integer parameter icut helps to fix this problem. The matrix of second derivatives is diagonalized and the first icut eigenvalues (these are the largest negative eigenvalues) are replaced by the icut+1st one. The thus modified eigenvalues and the eigenvectors are then used to build the working matrix of second derivatives. Note that the FBR matrix representation of { d^2/dx^2 - c/x^2) } is analytically evaluated ( c = -1/4, 0, 3/4, 2 for a = 1, 2, 3, 4). After this matrix is transformed to the DVR representation, the centrifugal term c/x^2 is removed by subtraction. The width parameter b has to be chosen carefully. Its optimal numerical value will depend on N, the number of grid points. Alternatively, one may use the input formats xi-xf (not recommended in general) or x0-xf. These formats compute b.

Remarks on sine DVR:

short: xi and xf denote, as usual, the first and last grid point. short is default and need not to be given.
long : xi and xf denote the boundaries of the sine basis functions (i.e. boundaries of the 'box') and not the first and last grid point.
Example: The following lines are equivalent.

  x   sin     19    1.00    19.0     short
  x   sin     19    0.00    20.0     long
  

  s1   sin  2  -0.5  0.5  spin
  

Remarks on cosine DVR:

The basis functions underlying the DVR are 1/sqrt(L), (2/sqrt(L))*cos[(j*pi/L)*(x-x₀)]. The symmetrized derivative, sdq = 0.5*( sin[(pi/L)*(x-x₀)] * d/dx + d/dx * sin[(pi/L)*(x-x₀)] ) is used as first derivative. Because only gerade (symmetric) wavefunctions are computed, we consider only the interval [x₀,x₀+L], although the wavefunction is periodic on the interval [x₀-L,x₀+L]. xi and xf are the first and last grid-point, but when long is given, the input is interpreted as x₀ and x₀+L.
The use of the operators dq, qdq, and cdq is not allowed for cos-DVR.
Example: The following lines are equivalent.

  x   cos     36    2pi/2
  x   cos     36    0.00      3.1415926535897  long
  x   cos     36    0.0436332313  3.097959422  short
  x   cos     36    0.0436332313  3.097959422

Remarks on FFT and exponential DVR:

FFT and exp-DVR have an identical set of input parameters (if exp-DVR is not combined with PLeg-DVR). These two methods are in fact largely equivalent and produce identical results (for same parameters). The numeric, however, is different and the exp-DVR will be faster for small grids whereas FFT is faster for long grids. Around 30 grid points both methods are of similar speed. The use of the interaction-picture is possible for exp-DVR.
FFT and exp-DVR enforce periodic boundary conditions, but they are often used for ordinary coordinates. A set of keywords adapts the input to the various situations:

linear: xi and xf are the coordinates of first and the last point of the grid. The grid-spacing is dx = (xf-xi)/(N-1). Due to the periodic boundary conditions the first grid point and the one following the last grid point are to be identified.

periodic: xi and xf are considered as identical due to the periodic boundary conditions. The grid-spacing is dx = (xf-xi)/N. The routine eingabe.f re-scales xf -> xf-dx.

s-periodic: xi and xf are considered as identical due to the periodic boundary conditions. The grid-spacing is dx = (xf-xi)/N. The grid points, however, are now placed symmetrically on the interval (xi,xf) and eingabe.f re-scales xi -> xi+dx/2, xf -> xf-dx/2.

2pi/mult or s-2pi/mult or c-2pi/mult: The routine eingabe.f sets xi=0 and xf=2*pi/mult and then performs according to the periodic or s-periodic keyword. For c-2pi/mult the grid is shifted such that xi=-xf.
Example: The following sets of lines are equivalent.

  x   fft     32    0.00       3.0434179    linear
  x   fft     32    0.00       3.1415927    periodic
  x   fft     32    2pi/2

or

  x   fft     32    0.0981748  6.1850105    linear
  x   fft     32    0.00       6.2831853    s-periodic
  x   fft     32    s-2pi

or

  x   fft     32   -3.0434179  3.0434179    linear
  x   fft     32   -3.1415927  3.1415927    s-periodic
  x   fft     32    c-2pi

Example:

For a system with two degrees of freedom, labeled X and Y, the following would define for X an FFT grid of 32 points from -2 to 2, and for Y a DVR basis of 32 harmonic oscillator functions generated with the given parameters. To show the use of the el-keyword we assume that there are three electronic states.

primitive-basis-section
  el     el     3
   X    FFT    32    -2.00    2.00    linear
   Y     HO    32     0.00    5.2,eV   1.00
end-primitive-basis-section

Remarks on External DVR:

Extern-DVR is an external DVR, where grid points and DVR derivative matrices are read from a file. The file can be read in ascii or binary format:

in ascii format (default):

do i=1,gdim
   read(unit,*) ort(g)
enddo
do i=1,gdim
   read(unit,*) (dif2mat(j,i),j=1,gdim)
enddo
do i=1,gdim
   read(unit,*) (dif1mat(j,i),j=1,gdim)
enddo

do i=1,gdim
   read(unit) ort(g)
enddo
do i=1,gdim
   read(unit) (dif2mat(j,i),j=1,gdim)
enddo
do i=1,gdim
   read(unit) (dif1mat(j,i),j=1,gdim)
enddo

where gdim is the basis_size. The file can have absolute or relative path. If file contains only grid points (for example in POTFIT program, when only grid points are used), the DVR matrices will be zeroed. If only second derivative matrix dif2mat is given, dif1mat will be zeroed.

Example:

     x   extern  30   x_data
     y   extern  50   y_data     binary
     z   extern  42   z_data     binary  angst
 theta   extern  23   theta_data deg

Remarks on KLeg, PLeg and sphFBR:

KLeg, PLeg and sphFBR all define 2D single-particle functions, although the basis definition is for each degree of freedom individually. KLeg must hence be followed by K and PLeg by exp and sphFBR by phiFBR.

The use if sphFBR is deprecated, PLeg or KLeg are preferred. WARNING: sphFBR does not work correctly if the potential is a potfit. Use PLeg instead. One may also consider to Fourier-transform the potential (with projection86) and then use KLeg. This approach is described e.g. in J.Chem.Phys. 123 (2005), 174311;   J.Chem.Phys. 128 (2008), 064305;   Mol.Phys. 110 (2012), 619-632. See also H2H2.inp and H2H2.op on the inputs or operatos directory, respectively

Example:

PRIMITIVE-BASIS-SECTION
  alpha   sphFBR   30   sym
  beta    phiFBR    5
  theta1  PLeg     31   even
  phi     exp      15   2pi   k=-6,6
  theta2  KLeg     31   even
  K_th     K       -5    5
end-primitive-basis-section

The KLeg/K, PLeg/exp and sphFBR/phiFB combinations generate mode-operators. Hence the DOFs (theta2, K_th), (theta1,phi), and (alpha,beta) must be combined with each other in the SPF-Basis-Section.

Remarks on Wigner DVR:

Wigner DVR defines 3D single-particle functions, although the basis definition is for each degree of freedom individually. The TWO degrees of freedom following Wigner are part of the combined mode and must be either K or exp, or any combination; i.e. Wigner/K/K, Wigner/K/exp, Wigner/exp/K, and Wigner/exp/exp are all valid choices. As these combinations generate mode operators, all three DOFs must be combined in the SPF-Basis-Section.

Important Note: The order of the three Wigner-DOFs must be   beta, gamma, alpha   or   beta, k, m   when k denotes the BF- and m the SF-component of the angular momentum. This ordering must hold in the PRIMITIVE-BASIS-SECTION, in the combination scheme defined in the SPF-BASIS-SECTION, and in the modes line of the HAMILTONIAN-SECTION. Note that the angles (beta,gamma,alpha) correspondent to the angular momenta (j,k,m), respectively.

Example:

PRIMITIVE-BASIS-SECTION
  ........
  beta    wigner  20   all
  gamma   exp     15   2pi
  alpha   k       -7   7
end-primitive-basis-section

SPF-BASIS-SECTION
  ........
  beta,gamma,alpha = 20
end-spf-basis-section
  

INIT_WF-SECTION

    file=geninwfrun                  file=geninwfrun
    select s1 m1 1 3 4               select  1 3 4
    select s1 m2 1 4 6 3             select m2 1 4 6 3
    select s2 m2 1 3 5	             select s2 m2 1 3 5 

After an initial WF is generated, either by file, build, read-inwf, or block-SPF / block-A / autoblock  this WF may be modified by one or several of the following procedures. The program will take actions in the order:
     A-coeff,  meigenf,   operate,    orthogonalise,    correction,   realpsi,
irrespectively of the order in the input file. Note that autoblock is executed in the propagation/relaxation step. Hence it is not meaningful to use operate or orthogonalise in connection with autoblock.

WARNING FORTRAN cannot open a file twice. Hence it may become necessary to copy a file such that the same information can be read via two different file-paths. Such a problem may arise when using block-SPF and block-A, or when using orthogonalise.

Building the initial wavefunction

The information needed to generate the initial wavefunction is written one line per degree of freedom between the keywords build and end-build. The input is free format, with blanks dividing the various parameters. The format for each line is

modelabel    type    parameters

The modelabel is an alphanumeric string attached to the degree of freedom. This must match a label specified in the primitive-basis section.

If one uses an electronic basis (i.e. one degree of freedom has the primitive basis type el), then the electronic initial state can be specified by the init_state keyword,

init_state = s

where the integer s specifies the initial state. This statement must be the only one on a line. If the lowest electronic state is selected, this keyword is not required.
Rather than init_state one may use euclid (see below) to specify the initial state. This becomes important, if there are several (single-set) electronic states.

To summarize:

When building the initial wavefunction, the type of the 1D functions can be chosen from one of the following:

The parameters needed for each type of function are as follows:

The initial single-particle functions are formed as follows:

• If there is an electronic degree of freedom treated in the single-set formulation, the single-particle functions are vectors representing the population of the states. The first vector is populated in the state specified by init_state, while the remaining vectors are generated orthonormal to this vector and fully populated in the remaining states. For example, if there are three states and the second state is initially populated, three vectors are produced, (0 1 0), (0 0 1) and (1 0 0), respectively.
• For the type gauss the first single particle function is a normalised Gaussian given an initial momentum momentum of the form f(x) = N exp(-1/4 ((x-centre)/width)²) exp(I momentum (x-centre)). Note: width may be chosen to be complex! (In this case the input reads "Re,Im,unit", e.g. "1.2,0.5,eV"). Higher single-particle functions are produced by multiplying the preceding function by x (so producing a series of powers of x), followed by Schmidt-orthogonalisation onto the lower functions.
• For the type HO the first single-particle function is a normalised harmonic oscillator of the form f(x) = N exp(-1/2 frequency mass (x-centre)²) exp(I momentum (x-centre)). Note: frequency may be chosen to be complex! (In this case the input reads "Re,Im,unit", e.g. "1.2,0.5,eV"). Higher single-particle functions are produced by multiplying the preceding function by x (so producing a series of powers of x), followed by Schmidt-orthogonalisation onto the lower functions.
• For the type HO odd the first single-particle function is a normalised harmonic oscillator of the form f(x) = N x exp(-1/2 frequency mass (x-centre)²) exp(I momentum (x-centre)). Note: frequency may be chosen to be complex! (In this case the input reads "Re,Im,unit", e.g. "1.2,0.5,eV"). Higher single-particle functions are produced by multiplying the preceding function by x^2, followed by Schmidt-orthogonalisation onto the lower functions.
• For the type HO even the first single-particle function is a normalised harmonic oscillator of the form f(x) = N exp(-1/2 frequency mass (x-centre)²) exp(I momentum (x-centre)). Note: frequency may be chosen to be complex! (In this case the input reads "Re,Im,unit", e.g. "1.2,0.5,eV"). Higher single-particle functions are produced by multiplying the preceding function by x^2, followed by Schmidt-orthogonalisation onto the lower functions.
• If the type Leg is chosen the single-particle functions are normalised (associated) Legendre polynomials. If the corresponding primitive basis is not Leg, then m must be zero. The parameter l defines which Legendre polynomial is taken as the first single-particle function. If the parameter symmetry equals nosym then all (even and odd l) Legendre polynomials are taken, and if sym, then only even (if l is even) or odd(if l is odd) Legendre polynomials are taken. Note the following restrictions: l >= m; if the corresponding primitive basis is Leg, then blz = l and ibleg = symmetry as defined in primitive-basis-section.
• In the sphFBR case the first single-particle function is a normalised spherical harmonic |j,m> = |j,m>. Higher single-particle functions are other spherical harmonics whose energies are as close as possible from the energy of the first one. Of course, the initial spherical harmonic must be part of the basis set.
• If eigenf is given, the single-particle functions are eigenfunctions of a one-dimensional Hamiltonian defined in the HAMILTONIAN-SECTION_XX of the operator file. The keyword pop selects which eigenfunction becomes the initial single-particle function. The ground state corresponds to pop=1, the first excited state to pop=2, and so on. If pop is set to a negative integer, only every second eigenfunction will be taken as initial single-particle function. This is useful, if the one-dimensional Hamiltonian to be diagonalised is symmetric. Thus, pop=-1 selects gerade eigenstates, and pop=-2 the ungerade ones. The selection (together with the computed eigenvalues) is protocolled in the log-file. If you want the eigenvectors too, specify "veigen" in the RUN-SECTION.
  Operators defined in a HAMILTONIAN-SECTION_XX are usually multi-dimensional operators. As a one-dimensional operator is required for diagonalisation, the program takes the uncorrelated part of the desired degree of freedom from the specified Hamiltonian. This is most conveniently described by an example. Assume that there are the DOFs x,y,z, and el. The eigenf-input reads:

   y  eigenf Hspf pop=2
  

  Then each of the following three Hamiltonians can be used equally well to generate the desired initial single-particle functions.

  HAMILTONIAN-SECTION_Hspf
  modes    |  y
  1.0      |  KE
  omy      |  q^2
  end-hamiltonian-section
  

  HAMILTONIAN-SECTION_Hspf
  ------------------------------------
  modes    |  x   |  y   |  z  |  el
  ------------------------------------
  1.0      |  1   |  KE  |  1  |  1
  omy      |  1   |  q^2 |  1  |  1
  ------------------------------------
  end-hamiltonian-section
  

  HAMILTONIAN-SECTION_Hspf
  ------------------------------
  modes    |  x   |  y   |  z
  ------------------------------
  1.0      |  KE  |  1   |  1
  omx      |  q^2 |  1   |  1
  1.0      |  1   |  KE  |  1
  omy      |  1   |  q^2 |  1
  1.0      |  1   |  1   |  KE
  omz      |  1   |  1   |  q^2
  const    | q^3  | sin  |  q
  ------------------------------
  end-hamiltonian-section
  

  The lines 1,2 and 5,6 of the tableau of the third Hamiltonian are ignored, as they refer to other degrees of freedom than the y one. The last line is ignored, as it represents a correlated Hamiltonian term.
  NB: There is no longer a need to set S1&1 for the electronic DOF, as it was the case in older versions. The old operator files, however, still work.
  eigenf   works for real and complex potentials, however the Hamilton to be diagonalized must be represented by a matrix. Hence eigenf does not work for FFT-DVR.
• If KLeg is chosen, the single-particle functions are normalised associated Legendre polynomials. The corresponding primitive basis must be KLeg or PLeg. The input line that follows must be for the type K. The parameter l defines first index (orbital angular momentum) of associated Legendre polynomial taken as the first single-particle function. The second (magnetic) index is given in the next K initial-wavefunction input-line. If the parameter symmetry equals nosym then all (even and odd l) Legendre polynomials are taken, and if sym, then only even (if l is even) or odd (if l is odd) Legendre polynomials are taken.
• The type K may appear after Leg, KLeg, gauss, or Wigner only. If the previous initial wavefunction is of type Leg or KLeg, the parameters here define the second (magnetic) index of the associated Legendre polynomials defined in the Leg or KLeg initial wavefunction. If the previous initial wavefunction is of type Wigner, the parameters here define the projections of angular momenta on the body-fixed and space-fixed z-axes for the first and second DOFs after Wigner, respectively, in the initial wavefunction. Then k is the initial value and kmin,kmax,dk are the bounds and the step of k. If the previous initial wavefunction is of type gauss then k indicates the initially occupied value of k-quantum number. In this case the last three parameters must be omitted. Note: The type K requires that the primitive basis must be KLeg and K or PLeg and exp or Wigner and exp/K and exp/K.
  The following example (inelastic N₂/LiF surface scattering) exemplifies the use of KLeg and K.

  PRIMITIVE-BASIS-SECTION
  ---------------------------------------------------
  # Mode    DVR      N      xi       xf
  ---------------------------------------------------
     z      fft     96   -0.75   4.5000  linear
     x      fft     24    0.00   5.3669  periodic
     y      fft     24    0.00   5.3669  periodic
     theta  PLeg    36    even
     phi    exp      9    2pi
  ---------------------------------------------------
  END-PRIMITIVE-BASIS-SECTION
  
  INIT_WF-SECTION
  build
  -----------------------------------------------------------
  # mode    type center  moment.  freq.         mass
  -----------------------------------------------------------
      z      HO   1.80   -24.d0 694.1683673,eV  1.d0
      x      HO   0.00     0.0    0.0           1.d0  # flat function
      y      HO   0.00     0.0    0.0           1.d0  # flat function
   theta     KLeg  2       sym      # j of initial spherical harmonic
     phi      K    1                # m of initial spherical harmonic
  -----------------------------------------------------------
  end-build
  end-init_wf-section
  

• If Wigner is chosen, the single-particle functions are normalized Wigner (small)-d functions. The corresponding primitive basis must be Wigner. The initial wave function must be K for the following TWO degrees of freedom. The j parameter defines the first index (total angular momentum) of the Wigner function taken as the first single particle function. The indices of the angular momenta about the body-fixed and space-fixed z-axes are given in the following two K-inwf lines. If the parameter symmetry equals nosym then all (even and odd j) Wigner functions are used; if sym, then only even or odd Wigner functions are used, depending on whether j is odd or even.
  Note that the angles (beta,gamma,alpha) correspondent to the angular momenta (j,k,m), respectively.
  The following table shows which combinations of primitive-basis and initial-wf types involving Wigner are allowed:

  For Wigner initial-wf:
  ---------------------------------------------
  DOF  init_wf    allowed primitive-basis types
  ---  ------- || -----------------------------
   1   wigner  || wigner only
  2,3  K only  || K or exp
  ---------------------------------------------
  
  For Wigner primitive-basis:
  ----------------------------------------------------------------------------------------
  DOF  p-basis  || allowed initial-wf types
  ---  -------  || -----------------------------------------------------------------------
   1   wigner   || wigner | gauss or leg           | any 1D type (HO,gauss,leg,eigenf,map)
  2,3  K or exp ||   K    | K (for p-basis=K only) | HO,gauss,eigenf,map (p-basis=exp only)
  ----------------------------------------------------------------------------------------
      

  An example involving the use of Wigner-inwf:

  PRIMITIVE-BASIS-SECTION
  ---------------------------------------------------
  # Mode    DVR      N  center  freq     mass
  ---------------------------------------------------
    r       HO       8  1.626   0.01837  1739.96
    beta    wigner  16    all
    gamma   exp     15    2pi
    m       k     -7  7
  ---------------------------------------------------
  END-PRIMITIVE-BASIS-SECTION
  
  INIT_WF-SECTION
  build
  -----------------------------------------------------------
  # mode    type  q.n. center  mom.  freq.    mass
  -----------------------------------------------------------
    r       HO         1.426   0.0   0.01837  1739.96
    beta   wigner 10   nosym excite=mkj  print  # initial-j, use odd and even j
    gamma    k     1  -2  2  1                  # initial-k, kmin, kmax, dk
    m        k    -2  -5  5  1                  # initial-m, mmin, mmax, dm
  -----------------------------------------------------------
  end-build
  END-INIT_WF-SECTION
      



• Mapping of initial (1D) single-particle functions is particularly useful when the initial SPF shall be generated by eigenf but an FFT is desired for the propagation step. The dummy grid may have fewer grid-points than the propagation grid, i.e the grid to which the SPF is mapped. However, the grid points of the dummy grid must coincide with those of the propagation grid. This, in practice, limits mapping to equidistant grids, i.e. FFT, exp-DVR and sin-DVR. The following example may be useful. Here the dummy grid coincides with the grid-points 6-65 of the propagation grid.

  PRIMITIVE-BASIS-SECTION
             .
             .
    R       FFT    192   -0.6          3.0
    R_dum   sin     60   -0.5057591623 0.6062827225
  end-primitive-basis-section
  
  INIT_WF-SECTION
  build
             .
             .
    R      map     R_dum
    R_dum  eigenf  operator    pop=1
  end-build
  end-init_wf-section
  

  Since the DOF   R_dum (in our example) is not defined in the SPF-BASIS-SECTION it will be ignored, when the operator is build. To avoid this, one has to use the addmode keyword.

  HAMILTONIAN-SECTION_operator
  addmode = R_dum
  ----------------------------------------
      modes     ......     | R_dum  | ....
  ----------------------------------------
             .
             .
  

  The map keyword can be used to sum up to three functions. A corresponding INIT_WF-SECTION may read:

  INIT_WF-SECTION
  build
   rd    map dum1  1.0
   rd    map dum2 -1.5
   rd    map dum3  (0.5,0.3)
   rv    HO     2.151  0.0  0.218,eV  13615.5
   theta gauss  2.22   0.0  0.0745
   dum1  gauss  4.310  0.0  0.075 pop=1
   dum2  gauss  4.315  0.0  0.080 pop=2
   dum3  gauss  4.320  0.0  0.085 pop=3
  

  The initial SPF for rd is a weighted sum of three harmonic oscillator functions. The first two coefficients are real, the third is complex. After summation, the function is normalized. The DOFs rd, dum1, dum2, and dum3 must be defined in the PRIMITIVE-BASIS-SECTION with identical grids.

Remarks on multi-set calculations:

For multi-set calculations, the initial functions can be defined differently for each state by using the state keyword:

modelabel    type    parameters    state = s

For multi-packet calculations, i.e. when packets > 1, the initial function definition must be given for each packet as

modelabel    type    parameters    pack = p

By default the 1st single particle function of each degree of freedom is used to form the initial Hartree product wavefunction. The initially populated single-particle function can be selected using the pop keyword:

modelabel    type    parameters    pop = i

Remarks on thermal relaxation calculations:

Which degrees of freedom are to be thermalized must be specified through the ran keyword in the INIT_WF-SECTION according to the order of parameters given in the table above labelled "parameter description". In the following we show four possible illustrative cases with an example. For the first mode the Ith state up to which the randomization will be averaged is not specified, thus the default value basis_size-1 will be employed. Meanwhile for the second mode, I is declared with a value of 30. Following this in the case of the third mode, the spf type eigenf is used together with the ran keyword. Finally, when mode-eigenfunctions are generated, the ran argument has to be included in the respective parameter's list line (meigenf keyword) as it is the case for the fourth mode. The meigenf procedure needs an initial SPF generated in the build-block. This initial SPF should not be an eigenfunction of the operator used in meigenf, but should have an overlap with several eigenfunctions. For this reason we have dispaced the origion of the HO-function by 0.25 au. Independently of the type of mode, the ran keyword will always be the last argument. Note that an electronic degree of freedom cannot be randomized. Electronic states are always considered as part of the system, not of a thermalized bath.

Example:

INIT_WF-SECTION
build
  init_state = 1
  q0001 HO     0.00  0.0  1.0 ran
  q0002 HO     0.00  0.0  1.0 ran=30
  q0003 eigenf oper    ran=30
  q0004 HO     0.25  0.0  1.0
end-build
meigenf = 4,oper,0,full,noselect,write,ran=30
end-init_wf-section
  

Reading the initial wavefunction

Note: Here all wavefunctions are assumed to be of multi-set format

Examples:
For a two state WF the following input is equivalent to a simple file = <restart-directory> statement.

Read-Inwf
  file = <restart-directory>
  orthopsi  # This is default, opposite: noorthopsi
  SPF 1 -> 1
  SPF 2 -> 2
  A   1 -> 1
  A   2 -> 2
end-read-inwf

Read-Inwf
  file = gs
  SPF 1 -> 1
  SPF 1 -> 2
  SPF 1 -> 3
  A   1 -> 2
end-read-inwf

Read-Inwf
  file = gs
  SPF 1 -> 1,2,3
  A   1 -> 2
end-read-inwf

Read-Inwf
  file = <restart-directory>
  SPF 1 -> 1,2
  init_state = 2
end-read-inwf

Read-Inwf
  file = run1
  SPF 1 -> 1
  file = run2
  SPF 1 -> 2
  A   1 -> 1
  file = run3
  A   2 -> 2
end-read-inwf

Read-Inwf
  file = <restart-directory>
  SPF 1 -> 1
  SPF 2 -> 2
  A   1 -> 1   weight = 0.65
  A   2 -> 2   weight = (2.3, -0.96)
end-read-inwf

Read-Inwf
  file = run1
  SPF 1 -> 2
end-read-inwf

Generating block initial wavepackets

INIT_WF-SECTION
 BUILD
  r1     gauss  1.20,Angst  0.0  0.030,Angst
  theta  gauss  3.14159     0.0  0.0524
  r2     gauss  1.25,Angst  0.0  0.025,Angst
 end-build

 A-coeff
  1 1 1 1 (1.d0,0.d0)
  2 1 1 2 (1.d0,0.d0)
  1 2 1 3 (1.d0,0.d0)
  1 1 2 4 (1.d0,0.d0)
 end-A-coeff
end-init_wf-section

INIT_WF-SECTION
 BUILD
  r1     gauss  1.20,Angst  0.0  0.030,Angst
  theta  gauss  3.14159     0.0  0.0524
  r2     gauss  1.25,Angst  0.0  0.025,Angst
 end-build
 autoblock
end-init_wf-section

INIT_WF-SECTION
 block-SPF = GS
 autoblock
end-init_wf-section

INIT_WF-SECTION
 block-SPF = GS/rst000,noorthopsi
 block-A   = GS/rst001,GS/rst003,GS/rst005
 block-A   = GS/rst007,GS/rst009,GS/rst011
end-init_wf-section

INTEGRATOR-SECTION

The string S=all after the integrator name may be omitted, i.e. ABM is equivalent to ABM/ALL.

For VMF calculations, only the BS, ABM or RK5/8 integrator may be used for the differential equations. Default is ABM. For VMF calculations the integrator must carry the extension S=all (or no extension at all), i.e. there is only one integrator within the VMF scheme.

For CMF calculations with standard (not multi-layer) MCTDH, the following combinations of integrators are possible: ABM/spf + SIL/A, BS/spf + SIL/A, RKx/spf + SIL/A, BS/all, ABM/all, RKx/all. Default is BS/spf + SIL/A.

For CMF calculations with multi-layer MCTDH, there are two different schemes of integrating the equations of motion. The scheme can be selected by the keyword ml-cmf (see below). For unified propagations, the complete EoM is integrated as one, hence the choice of integrator is restricted to ABM/all, BS/all, or RKx/all. (For legacy reasons, here /spf has the same effect as /all.) For split propagations, the EoM for each mode is integrated by itself, hence for each mode one can choose its individual integrator/parameters. (Note: Currently the choice of integration algorithm is restricted to RK5/8 or (C)SIL; (C)SIL only makes sense for mode 1.) This is done by using S=modespec, where the mode specification modespec is one of the following:

The modes are numbered consecutively according to the ML-BASIS-SECTION. One can double-check the mode numbers by inspecting the visualization of the ML tree generated by the graphviz keyword [RUN-SECTION]. For example inputs see test38.inp and test39.inp which are on $MCTDH_DIR/elk_inputs.

For numerically exact calculations (i.e. the keyword exact has been specified in the RUN-SECTION), any of the integrators may be chosen, with S = all, or S = spf. Default is ABM, but more efficient is usually SIL (or CSIL).

The Davidson "integrator", DAV, (actually a diagonaliser, of course) is the optimal choice for the "A-propagation" of improved relaxation. The accuracy parameter is an upper limit (in au) for the error of the eigenvalue. Improved relaxation requires CMF/varphi or CMF/fix. (Simply use CMF). The routine DAV is for hermitian Hamiltonians and general wavefunctions. rDAV is for real Hamiltonians (i.e. H*psi = real for all real psi) and real (initial) WF. Memory is saved as the Davidson vectors are stored as real. Most of the arithmetic (in particular H*psi), however, is still complex. This problem is partly solved by rrDAV. rrDAV uses the same Davidson routine, but a simplified routine is employed to perform the matrix product H*A. This routine is faster, because it uses more real arithmetic, but works only for simple Hamiltonians (only real operators, e.g. no p). NB: The propagation of the SPFs is always done in complex arithmetic. cDAV is for non-hermitian Hamiltonians. This allows to compute resonances of CAP augmented Hamiltonians. The convergence, however, is slower. cDAV is chosen automatically, if CAPs are present.
The Davidson routines DAV, rDAV, and rrDAV can be used to perform block-relaxations. To this end set the packets keyword in the SPF-Basis-Section. Note that cDAV cannot be used in block form. All Davidson routines allow for parallelization (pthreads and/or mpi).

NOTE: There are two Lanczos integrators, a real version for hermitian Hamiltonians and a complex version (Lanczos-Arnoldi) for non-hermitian ones. If one gives the keyword SIL, then the program tries to detect whether the Hamiltonian is hermitian or not and it makes the appropriate choice. This, however, works safely only for CAPs. When one knows that the Hamiltonian is non-hermitian, one should use CSIL. The use of SIL in case of a non-hermitian Hamiltonian may lead to wrong results.

Omitted integrator parameters default to:

ALLOC-SECTION

Example Inputs for wave functions

• NOCl S0 (long)
• NOCl S0 (short)
• Relaxation on NOCl S0 surface
• Propagation on NOCl S1 surface
• CH₃I (relaxation, single-set)
• CH₃I (propagation, single-set)
• CH₃I (relaxation, multi-set)
• CH₃I (propagation, multi-set)
• H+D₂ reactive scattering (Coupled States Approximation)
• H+D₂ reactive scattering
• Pyrazine (6 modes)
• N₂-LiF surface scattering
• 1D harmonic oscillator (diagonalisation)
• 2D LiCN model (diagonalisation)
• CO₂ vibrational spectrum
• Further example inputs

Example Inputs for use of CPP with MCTDH

• Allene E state vibrational spectrum
• Allene B state vibrational spectrum

• Include file #1 with common settings
• Include file #2 with common settings