operators (expect) Real space densities -3D first order real density (rho) -3D laplacian of the real electron density (lap) -3D B functions (bfun) -3D ELF localization function (elf) ..... .. T O P O L O G I E S... For all the 3D and 1D scalars, not only their magnitudes, but also their first and second derivatives (gradients and hessians) are computed analytically. This allows us to make automatic topological analyses of of them. A Morse consistency check is done on all topologies, and paths connecting (3,-1) to (3,-3) points are traced and studied. ...... .. G R I D S .... 0D, 1D, 2D and 3D grids are allowed for all the scalars separately, in general format. 3D cube (gaussianXX format) multigrids are also allowed, so isosurfaces of the scalars are easily displayed by standard molecular graphics programs like molekel. ...... .. A S S O C I A T E D S C A L A R S ..... A number of interesting associated scalars related to the real space density are also obtained. These include: -kinetic energy densities: K (positive definite) G (Schrodinger density) P (Pauli repulsion kinetic energy density) -kinetic stress tensor -Potential stress tensor (Maxwell selfinteraction-less form) -Ehrenfest Force densities (~ eigenvalues of the kinetic stress tensor) -Potential energy densities: nn (nucleus-nucleus) ne (electron-nucleus) ee (electron-electron) !Monodeterminantal (Beware that it is also computed in non- determinantal, meaningless cases) hf (Hartree-Fock total energy density) -Nuclear and electronic electric fields -nuclear and electronic total force densities (f+(r), f-(r)) ....... .. B A S I N I N T E G R A T I O N As of version 1.00, promolden includes a variety of methods to integrate the above real density associated scalars within Bader atomic basins. In the near future, elf integration will also be available. The user is warned about the computational efficiency of these integrations. It is extremely expensive to integrate maxwell densities numerically. Both search-of-basin-boundary and Biegler-Konig's natural-coordinate methods have been implemented. Several adaptive and non-adaptive integration quadratures are also allowed. ..... ---------------------------------------------------------------------- ---------------------------------------------------------------------- INPUT The code is invoked: promolden input.file [output.file] The input is keyword oriented, freeformat, and the output is meant to be selfcontained. -The first line must contain the name of the wavefunction file in the AIMPAC WFN format. We have finally not included other wavefunction formats, but they are easily added if needed. Notice that the precision of the primitive coefficients on building molecular orbitals may be smaller than that needed for very, very accurate momentum density properties. The latter are known to be extremely dependent on the former. -The following lines are not mandatory. Each of them is an order for promolden, and the user may concatenate as many as he/she desires. All of them will be executed in the order given. They consist of a keyword name and numeric/character options. There are 4 general types of keywords: -informative keywords : Activate general options of the program. -scalar keywords : Obtain scalars (sometimes on grids). -topology keywords : Demand the topology of a scalar field. -basin integ. keywords: Integrate rho associated scalars in atomic basins -A description of each keyword and its options follows: INFORMATIVE KEYWORDS: ----------- NOMAXWELL: By default, all maxwellian stress tensors are computed on spatial density calculations. As their computation is time consuming, the user may force NOT to obtain them. NOBFUNC : Similar arguments apply to the calculation of B functions in cube grids. (see below). None of these keywords admit additional parameters. SCALAR KEYWORDS: ---------------- PI, RHO, ELF, FORM, BFUN: Demand the computation of the scalars in 0D,1D,2D,or 3D grids. On output, a record per point is printed in a format suitable for most graphing codes, like Gnuplot. This record has the following form: x,y,z,scalar,(grad(i),i=1,3),((hess(i,j),j=1,3),i=1,3), laplacian. i runs over x,y,z. Keyword=Pi,Elf,Form,Bfun In the case of the spatial density rho, the number of properties per record is greater: x,y,z,scalar,(grad(i),i=1,3),((hess(i,j),j=1,3),i=1,3), laplacian,g,k,p,(-dsig(i),i=1,3),[nn,ne,ee,(fielde(i),i=1,3), (fieldn(i),i=1,3),elf,(f+(i),i=1,3),(f-(i),i=1,3)] Here, g,k,p are the kinetic energy densities; nn,ne,ee the maxwellian potential energy densities; fielde/n the electronic/nuclear components of the electric field; f+/- the nuclear/electronic force densities. All magnitudes within brackets [] are not computed if the NOMAXWELL flag is turned on. PIP, RHOP, ELFP, FORMP, BFUNP: Demand the computation of the scalars in 0D,1D,2D,or 3D grids. On output, a set of records is produced at each point of the grid, including diagonalization of the hessian matrix. These kind of output is intended to be more easily read by humans. All the previous properties are computed and displayed in a common way. Usage: Each of these keyword is followed by the following parameters. KEYWORD ndim xvec0 [xvec1 [xvec2 [xvec3]] npoints ] A ndim-dimensional GRID will be computed. 0<=ndim<=3. xvecn, 0<=n<=2, are points in real or momentum space defining a ndim-dimensional parallelepiped. |xvec3 | | | / --------- xvec2 / xvec0 / xvec1 ndim=0 properties at a point xvec0 ndim=1 properties along a line defined by the xvec0 and xvec1 points. Npoints will be computed. ndim=2 properties in the rectangle defined by the xvec0, xvec1, xvec2 points. Npoints X Npoints in the grid ndim=3 properties in the box defined by the xvec0, xvec1, xvec2, xvec3 points. Npoints X Npoints X Npoints in the grid EXPECTATION: Obtain the expectation values of the

^n operators for the molecule under study. -2<=n<=4 SPI : Obtain the spherically averaged momentum density. Usage: SPI pmax, npoints This order construct a curve of the spherically averaged momentum density from p=0 to p=pmax using npoints. CUBE : Obtain a "cube" file (to be used by molecular graphics programs) for a set of relevant scalars. A file named as the wavefunction filename plus an informative suffix will be written for each scalar. These files are immediately read by MOLEKEL. If NOMAXWELL or NOBFUNC flags are active, no maxwellian nor B function files will be written. Usage: CUBE x, y, z, nx, ny, nz This order will generate cube files centered in the box -x/2<=x<=x/2, -y/2<=y<=y/2, -z/2<=z<=z/2 using nx, ny, and nz points in each direction. At the moment k,g,nn,ne,ee,vt,et,elf,rho,lap,grad,b, blap,bgrad files will be written. All of them have their common meaning. vt is the total potential energy density; et is the total energy (HF) density; grad is the gradient module; b,blap, bgrad are the B function, and its laplacian and gradient module. SQUARE :Obtain a proaim "grid" file to be used directly by the proaim codes "contor" and "relief" to make isoline and surface plots of a set of relevant scalars in a plane. A file named as the wavefunction filename plus an informative suffix will be written for each scalar. If NOMAXWELL or NOBFUNC flags are active, no maxwellian nor B function files will be written. Usage: SQUARE xvec0, xvec1, xvec2, nx, ny This order will generate grid files on a plane described using the same notation as in the scalar keywords examples given above. xvec0, xvec1, and xvec2 are 3D vectors. nx X ny grid points will be used. At the moment k,g,nn,ne,ee,vt,et,elf,rho,lap,grad,b, blap,bgrad files will be written. See the CUBE keyword for more information. TOPOLOGY KEYWORDS: ------------------ TPI,TRHO,TLAP,TELF,TBFUN: Obtain automatically the topology of the 3D scalars PI, RHO, LAP, ELF, and BFUN, respectively. It has been demonstrated over the years that a simple Newton-Raphson (NR) search of the zeros of the equation: -> -> Grad(Scalar)=0, is, probably, the simplest and quickest way of obtaining the complete topology of these scalar densities. For each scalar, a full search is made by starting NR steps at points uniformly distributed over a sphere of a given maximum radius and given number of points along the r, theta, and phi directions. A judicious value of the maximum radius of the sphere and the number of trial points gives almost invariably the complete topology of the scalar. A wealth of information is given at each critical point found. A Morse analysis is done on the final topology, and the (3,-1) to (3,-3) gradient paths are computed. Usage: KEYWORD rmax, np, epsilon Obtain the topology within a sphere of radius rmax, using np X np X np radial,theta, and phi points, and accepting a critical point when |Grad(scalar)|