# Properties and expectation values

## The property program

The property program allows the evaluation of one-electron operators and expectation values. Normally, the operators are computed automatically when using the global `GEXPEC`

directive (see section One-electron operators and expectation values (GEXPEC)) or the `EXPEC`

or `TRAN`

commands in the SCF, MCSCF, and CI programs. The explicit use of the property program is only necessary in the rare case that the user is interested in an orbital analysis of the properties.

### Calling the property program (PROPERTY)

`PROPERTY`

invokes the property program.

### Expectation values (DENSITY)

`DENSITY`

[,*record.file*] [,*specifications*]

If this card is present, the density matrix will be read from record *record.file* and property expectation values will be calculated. If the specification *record.file* is omitted, the last dump record is used. Density matrices for specific states can be selected using *specifications*, as explained in section selecting orbitals and density matrices (ORBITAL, DENSITY). Note that the density matrices are stored in the same record as the orbitals.

### Orbital analysis (ORBITAL)

`ORBITAL`

[,*record.file*] [,*specifications*]

If this card is present, the orbitals are read from record *record.file* and an orbital analysis of the expectation values is printed (the density matrix must also be provided!). If *record.file* is omitted, the last dump record is used. This is only meaningful for diagonal density matrices (SCF or natural orbitals). Natural orbitals for specific states can be selected using *specifications*, as explained in section selecting orbitals and density matrices (ORBITAL, DENSITY).

### Specification of one-electron operators

The required operators are specified by code words. Optionally, the geometry or the nuclear centre at which the operator is computed can be specified.

For each operator, an input card of the following form is required:

*code,centre,x,y,z,,factor*

*code* specifies the property. The available operators are given in section One-electron operators and expectation values (GEXPEC).

The other parameters have the following meaning:

row number of Z–matrix or atomic symbol defining the centre at which property shall be calculated; if*centre**centre*$\ne 0$ you need not read in coordinates.cartesian coordinates of the point (only if*x,y,z**centre*=0).the operator is multiplied by this factor. The default is*factor**factor*=1 except for`REL`

. In this cases proper factors for relativistic corrections are used unless*factor*is given. The two commas before factor are needed to preserve compatibility with`Molpro96`

.

### Printing options

`PRINT`

,*print*

This card is used to control output, mainly for debugging purposes.

no test output (default)*print*$= 0$operators are printed.*print*$> 0$

### Examples

The following example computes the dipole quadrupole moments of water and prints an orbital analysis. By default, the origin is at the centre of mass, and this is taken as origin for the quadrupole moments.

- examples/h2o_property.inp
***,h2o properties geometry={o;h1,o,r;h2,o,r,h1,theta} !Z-matrix geometry input r=1 ang !bond length theta=104 !bond angle hf !do scf calculation property !call property program orbital !read scf orbitals density !read scf density matrix dm !compute dipole moments and print orbital contributions qm !compute quadrupole moments and print orbital contributions {multi;wf;state,2;dm !do full-valence CASSCF natorb,state=1.1 !compute natural orbitals for state 1.1 natorb,state=2.1} !compute natural orbitals for state 2.1 {property !call property program orbital,state=1.1 !read casscf natural orbitals for state 1.1 density,state=1.1 !read casscf density matrix for state 1.1 dm !compute dipole moments and print orbital contributions qm} !compute quadrupole moments and print orbital contributions {property !call property program orbital,state=2.1 !read casscf natural orbitals for state 2.1 density,state=2.1 !read casscf density matrix for state 2.1 dm !compute dipole moments and print orbital contributions qm} !compute quadrupole moments and print orbital contributions

Alternatively, the dipole and quadrupole moments can be computed directly in the SCF and MCSCF programs, but in this case no orbital contributions are printed:

- examples/h2o_gexpec1.inp
***,h2o properties geometry={o;h1,o,r;h2,o,r,h1,theta} !Z-matrix geometry input r=1 ang !bond length theta=104 !bond angle gexpec,dm,qm !global request of dipole and quadrupole moments hf !do scf calculation {multi;wf;state,2 !do full-valence CASSCF natorb,state=1.1 !compute natural orbitals for state 1.1 natorb,state=2.1} !compute natural orbitals for state 2.1

There is a another example described in the Program Control section, as well as:

- examples/h2o_prop.inp
***,H2O properties r=1.85 !define initial bond distance theta=104 !define initial bond angle (=2 theta) geometry={o;h1,o,r;h2,o,r,h1,theta} !geometry input Basis={ s,1,v;c,1.6; !van Duijneveld 13s, first 6 contracted p,1,v;c,1.4; !van Duijneveld 8p, first 4 contracted d,1,3d;f,1,2f !Dunning optimized 3d, 2f s,2,v;c,1.4;p,2,3p} !8s3p basis for H gexpec,qm,ef1,ef2,fg,,0,0,0.5 !Field gradient at (0,0,0.5) hf !closed-shell SCF pop !Mulliken population analysis using scf density {dma !distributed multipole analysis using scf density limit,,3}; !limit to rank 3 multi !full valence CASSCF {pop; !Mulliken population analysis using mcscf density individual} !give occupations of individual basis functions {dma; !distributed multipole analysis using mcscf density limit,,3} !limit to rank 3 {ci !MRCI with casscf reference natorb,2350.2 dm,2350.2} !save MRCI density matrix on record 2350, file 2 {pop;density,2350.2; !Mulliken population analysis using MRCI density individual} !give occupations of individual basis functions {property; density,2350.2; !use property program to compute expectation values orbital,2350.2; qm;ef,1;ef,2;fg,,,,0.5} !same operators as before ---

## Distributed multipole analysis

Any density matrix can be analysed using the distributed multipole analysis described by Stone, Chem. Phys. Lett. **83**, 233 (1981). The multipole moments arising from the overlap of each pair of primitives are calculated with respect to the overlap centre, and then shifted to the nearest of a number of *multipole sites*. By default these comprise all atoms specified in the integral input. However the list of multipole sites can be modified by deleting and/or adding sites, and also by restricting the rank of multipole which may be transferred to any given site. The atomic charges are stored in the `MOLPRO`

variable `ATCHARGE`

. The i’th element in `ATCHARGE`

corresponds to the i’th row of the Z-matrix input.

Options may appear in any order, except `DENSITY`

, which must be first if given.

The present version does not allow generally contracted AO basis sets.

### Calling the DMA program (DMA)

`DMA;`

This command initializes the DMA program.

### Specifying the density matrix (DENSITY)

`DENSITY`

,*record.file* [,*specifications*]

The density matrix to be analysed is that found in record *record* on file *file*. If omitted, *record.file* defaults to current orbital record. If specified, `DENSITY`

must appear first in the input. Density matrices for specific states can be selected using *specifications*, as explained in section selecting orbitals and density matrices (ORBITAL, DENSITY).

### Linear molecules (LINEAR, GENERAL)

`GENERAL`

;

(default) invokes the normal program, which copes with any geometry.

`LINEAR`

invokes a faster program which can be used when all the atoms are arranged parallel to the $z$-axis and only the $m=0$ components of the multipoles are required.

### Maximum rank of multipoles (LIMIT)

`LIMIT`

,*name,lmax*;

*lmax* is the highest rank of multipole that is to be calculated by the program. Default (and maximum) is 10 for the general program and 20 for the linear one. If *name* is specified, the limit applies only to multipole site *name*.

### Omitting nuclear contributions (NONUCLEAR)

`NONUCLEAR`

The nuclear contributions to properties are not to be evaluated.

### Specification of multipole sites (ADD, DELETE)

`ADD`

,*name,x,y,z,lmax,radius*;

Add a new site at ($x$, $y$, $z$) with the *name* specified. The multipole rank is limited to *lmax* if a value is specified, otherwise the value of lmax specified by the `LIMIT`

directive is used. No account is taken of symmetry; every site in a symmetry-equivalent set must be specified explicitly. The *radius* of the site may also be specified (default 1.0).

`DELETE`

,*name*

Delete all atoms with the *name* given from consideration as a multipole site. Note that original atoms from the integral program have names $1,2,3,\ldots$ as printed in integral output. `DELETE,ALL`

deletes all atoms and gives the multipoles with respect to the origin only.

### Defining the radius of multipole sites (RADIUS)

`RADIUS`

,*name,r*;

Assign radius $r$ to all sites with the *name* given. The program moves multipoles at an overlap centre $P$ to the site $S$ for which the value of $|P-S|/r(S)$ is smallest. In the absence of a `RADIUS`

directive, all sites are given radius 1.

### Notes and references

The multipoles produced by this analysis are given in their spherical harmonic definitions. Explicit formulae for translating between the cartesian and spherical harmonic definitions of the multipole moments are given in, *Explicit formulae for the electrostatic energy, forces and torques between a pair of molecules of arbitrary symmetry*, S. L. Price, A. J. Stone, and M. Alderton, Molec. Phys., 52, 987 (1984).

For examples of the use of `DMA`

analysis see, Price and Stone, Chem. Phys. Lett. **98**, 419 (1983); Buckingham and Fowler, J. Chem. Phys. **79**, 6426 (1983).

### Examples

The following input calculates SCF multipole moments for water.

- examples/h2o_dma.inp
***,h2o distributed multipole analysis geometry={o;h1,o,r;h2,o,r,h1,theta} !Z-matrix geometry input r=1 ang !bond length theta=104 !bond angle basis=6-311g** hf !do scf calculation {dma;limit,,4} !results for total multipoles are

## Mulliken population analysis

### Calling the population analysis program (POP)

`POP`

;

Invokes Mulliken analysis program, which analyses any density matrix into its contributions from s,p,d,f… basis functions on each atom. In the case of Hartree-Fock, density functional, and MCSCF calculations, the density matrix is taken from the last dump record, unless overridden with the `DENSITY`

card. For other methods, the density matrix has to be written and then must be referred to with the `DENSITY`

card.

The subcommands may be abbreviated by the first four characters. The atomic charges are stored in the `MOLPRO`

variable `ATCHARGE`

. The i’th element in `ATCHARGE`

corresponds to the i’th row of the Z-matrix input.

### Defining the density matrix (DENSITY)

`DENSITY`

,*record.file* [,*specifications*]

Take density matrix to be analysed from record *record* on file *file*. Density matrices for specific states can be selected using *specifications*, as explained in section selecting orbitals and density matrices (ORBITAL, DENSITY). Note that the density matrices are stored in the same record as the orbitals.

### Populations of basis functions (INDIVIDUAL)

`INDIVIDUAL`

;

### Examples

- examples/h2o_pop.inp
***,h2o population analysis geometry={o;h1,o,r;h2,o,r,h1,theta} !Z-matrix geometry input r=1 ang !bond length theta=104 !bond angle basis=6-311g** hf !do scf calculation pop; !Mulliken population analysis using mcscf density individual !give occupations of individual basis functions

If specified, the Mulliken populations of each individual basis function are printed.

The following example shows how to compute the population first on the Hartree-Fock level, and then on the CI level.

- examples/ch4_ci_pop.inp
geomtyp=xyz; geometry={ c -1.058493 -0.384055 0.000000 h -1.058493 0.695337 0.000000 h -0.040825 -0.743829 0.000000 h -1.567327 -0.743829 0.881326 h -1.567327 -0.743829 -0.881326 } basis={ default=def2-SVP } hf; pop; ci;dm,9000.2; pop;density,9000.2

## Natural Bond Orbital Analysis

### Calling the Natural Bond Orbital analysis program (NBO)

`NBO`

,[`WITH_CORE=`

*core_option*],[`LEVEL=`

*level*],[`KEEP_WBI=`

*wbi_option*];

The Natural Bond Orbital Analysis of Weinhold and coworkers (J. Chem. Phys. **83**, 735 (1985), J. Chem. Phys. **83**, 1736 (1985) and J. Chem. Phys. 78 (1983) 4066) can be called by the use of the `NBO`

card. It reads from a density or orbital record, and performs the necessary transformations to Natural Atomic Orbitals (NAO), Natural Bond Orbitals (NBO) and Natural Localized Molecular Orbitals (NLMO). The latter can also be saved to a record and later used in local correlation treatments (cf. Section PAO-based local correlation treatments). By default, the full orbital space is used. The core orbitals can, however, be left out of the procedure if *core_option*=0.

One can choose to truncate the transformation series (e.g., only compute the NAO orbitals), with help of the `LEVEL`

keyword. If *level*=1, only the NAO transformation will be carried out. For *level*=2 the NBO transformation is performed, and for 3 the NLMO (default).

Sometimes, the NBO procedure will not converge due to a bad ordering on the 2-center bond search. The first run is based on the Wiberg bond index, but the algorithm switches to the atom ordering on the subsequent runs. This can be avoided by the use of the option `KEEP_WBI`

. If *wbi_option*=1, the Wiberg bond index is used in all iterations.

### Saving the NLMO orbitals (SAVE)

`SAVE,`

*record.file*;

The NLMO orbitals are saved in the specified *record*, together with the NPA charges.

## Finite field calculations

Dipole moments, quadrupole moments etc. and the corresponding polarizabilities can be obtained as energy derivatives by the finite difference approximation. This is most easily done with the `DIP`

, `QUAD`

, or `FIELD`

commands, which add a specified amount of the dipole, quadrupole, or any general, operators to the Hamiltonian. An error will result if the added perturbation is not totally symmetric (symmetry 1). Note that the orbitals must be recomputed before performing a correlation calculation.

### Dipole fields (DIP)

`DIP`

,*dx,dy,dz*;

`DIP+`

,*dx,dy,dz*;

Add a finite combination of the dipole operators $\vec\mu=(\mu_x, \mu_y, \mu_z)$, $H_1=\vec d \cdot \vec \mu$ ( $\vec d=(\textit{dx},\textit{dy},\textit{dz})$) to the Hamiltonian (both the 1-electron operator and the core energy). `DIP+`

adds to any existing field, otherwise any previous perturbation is removed.

The perturbed hamiltonian represents a physical system in a uniform electric field with electric field strength $\vec F= -\vec d$. Therefore the corresponding energy-derivative form of the dipole moment projection in this direction can be obtained as $$|\vec F|^{-1}\vec F \cdot \vec \mu = |2\vec d|^{-1}(E(\vec d)-E(-\vec d)) + O(|\vec d|^2)= |\vec d|^{-1}(E(\vec d)-E(\vec 0)) + O(|\vec d|).$$ The diagonal polarisability in this direction can similarly be calculated via $$\alpha_{\vec d, \vec d} = |\vec d|^{-2}(E(\vec d)+E(-\vec d)-2E(\vec 0)) + O(|\vec d|^2).$$

### Quadrupole fields (QUAD)

`QUAD`

,*qxx,qyy,qzz,qxy,qxz,qyz*;

`QUAD+`

,*qxx,qyy,qzz,qxy,qxz,qyz*;

Exactly as the `DIP`

command, but adds a combination of quadrupole operators to the Hamiltonian.

### General fields (FIELD)

`FIELD`

,*oper1,fac1, oper2,fac2, …*;

`FIELD+`

,*oper1,fac1, oper2,fac2, …*;

Adds one-electron operators *oper1*, *oper2*, …with the corresponding factors *fac1*, *fac2*, …to the one-electron hamiltonian. The available operators are given in section One-electron operators and expectation values (GEXPEC). An error will result if the added perturbation is not totally symmetric (symmetry 1).

`FIELD+`

adds to any existing field, otherwise any previous field is removed.

Note that `FIELD`

does currently not modify core polarization potentials (CPP). If CPPs are present, only `DIP`

and `QUAD`

should be used.

### Examples

The first examples shows various possibilities to add perturbations to the one-electron hamiltonian.

- examples/field.inp
***,H2O finite fields R = 0.96488518 ANG THETA = 101.90140469 geometry={H1 O,H1,R; H2,O,R,H1,THETA} {hf;wf,10,1} !scf without field f=0.05 dip,,,f !add dipole (z) field to h0 hf !do scf with modified h0 field,dmz,f !add dipole (z) field to H0 !same result as previous example hf !do scf with modified h0 quad,,,f !add quadrupole (qmzz) field to h0 hf !do scf with modified h0 field,qmzz,f !add quadrupole (qmzz) field to h0; !same result as previous example hf !do scf with modified h0 field,zz,f,xx,-0.5*f,yy,-0.5*f !add general field; same result as quad above hf !do scf with modified h0 field,zz,f !same as before with separate field commands field+,xx,-0.5*f field+,yy,-0.5*f hf !do scf with modified h0 field !remove field hf !scf without field

The second example shows how to compute dipole moments and polarizabilities using finite fields.

- examples/h2o_field.inp
***,H2O finite field calculations r=1.85,theta=104 !set geometry parameters geometry={O; !z-matrix input H1,O,r; H2,O,r,H1,theta} basis=avtz !define default basis field=[0,0.005,-0.005] !define finite field strengths $method=[hf,mp4,ccsd(t),casscf,mrci] k=0 do i=1,#field !loop over fields dip,,,field(i) !add finite field to H do m=1,#method !loop over methods k=k+1 $method(m) !calculate energy e(k)=energy !save energy enddo enddo k=0 n=#method do m=1,#method k=k+1 energ(m)=e(k) dipmz(m)=(e(k+n)-e(k+2*n))/(field(2)-field(3)) !dipole moment as first energy derivative dpolz(m)=(e(k+n)+e(k+2*n)-2*e(k))/((field(2)-field(1))*(field(3)-field(1))) !polarizability as second der. enddo table,method,energ,dipmz,dpolz title,results for H2O, r=$R, theta=$theta, basis=$basis ---

## Relativistic corrections

Relativistic corrections may be calculated within the Cowan-Griffin approach by computing expectation values of the mass-velocity and 1-electron Darwin integrals; these should be generated using the property integral program with keyword `REL`

The expectation values can be computed within the SCF, MCSCF and CI programs in the usual way using the `EXPEC`

command, again with the keyword `REL`

. The mass-velocity and Darwin terms, and their sum are subsequently available through the Molpro variables `MASSV`

, `DARWIN`

and `EREL`

respectively. An example can be found here.