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AMPAC 8 User Manual |
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AMPAC 8 User Manual |
AMPAC™8 offers users a vastly improved and expanded configuration interaction (CI) capability when compared to other semiempirical programs. Given the special keywords and preselection techniques that have been implemented, AMPAC can now perform both extremely complex, yet efficient CI calculations.
The general-purpose semiempirical models available in AMPAC were originally developed for the efficient prediction of ground state molecular properties at equilibrium geometries. This is where the models were parameterized and where the vast majority of data exists. The level of theory upon which these models are based is the Hartree-Fock SCF method, restricted or unrestricted, which has proved sufficiently accurate to reproduce and predict ground-state properties of most "ordinary" organic molecules and many molecules containing only main-group elements. For excited state geometries and properties, some open-shell systems, atoms or molecules containing metals (e.g., transition metal complexes), properties which involved excited states (e.g., UV/visible spectra) and unusual geometries (e.g., transition states corresponding to weakly avoided crossings) the Hartree-Fock level of theory is often inadequate at best. The need for efficient semiempirical methods capable of handling these cases has long been recognized, especially for larger molecules. Dedicated methods parameterized specifically for some of these cases (e.g., UV/visible spectra) have been proposed and implemented. In AMPAC, the approach taken is to use a post-SCF configuration interaction (CI) method, starting with a semi-empirical SCF wavefunction already available. In semiempirical models, some of the "non-specific" electron correlation is absorbed by the parameterization and a crude large-scale CI is usually meaningless. Therefore, an "adapted" CI capable of modeling primarily the "specific" electron correlation necessary to describe excited states and open-shells systems is usually preferable.
An AMPAC N-electron wavefunction Ψ is always initially defined to be a single-determinant SCF wavefunction ΨSCF, i.e., a determinant ψ of orthonormal, variationally determined spin orbitals (SOs) χ. For RHF, the spatial components (MOs) φ of the SOs are restricted to be identical in pairs of alpha and beta electrons (or "half-electrons" of fractional charge for open-shell RHF) during the SCF, whereas this is not the case in UHF. In some RHF cases, it is desirable or necessary (e.g., open-shell RHF, calculation of UV/visible spectra, etc.) to go beyond SCF to a more sophisticated Configuration Interaction (CI) method (CI cannot be used with UHF), where Ψ = ΨCI is one of many possible variationally determined linear combinations of determinants ψ. The set of determinants ψ combined to form ΨCI is called an N-electron basis (in contrast, to the one-electron basis of Slater orbitals used to expand the SOs) and includes a reference determinant ΨRef (usually ΨSCF) as well as a set of "excited" determinants obtained by moving one or more of the electrons from the occupied SOs of ΨRef to corresponding virtual SOs. For open-shell RHF, the reference wavefunction ΨRef is not ΨSCF since it contains fractionally occupied SOs, but rather a determinant obtained by filling the SOs of ΨSCF in the standard way using the Aufbau principle. In AMPAC, a single determinant ψ is referred to as a "microstate". A general expression for ΨCI can be given by:
where the "C"s are the linear coefficients to be determined, ψia is the determinant resulting from moving an electron from occupied SO χi to virtual SO χa, ψia,jb is the determinant resulting from moving electrons from occupied SOs χi and χj to virtual SOs χa and χb respectively, etc. The sums involving i, j, k, ... are over some subset of occupied SOs in ΨRef while the sums involving a,b,c,... are over some subset of virtual SOs in ΨRef. Together, these SOs define the CI-active MOs, or the "active space". A set of CI wavefunctions and corresponding energies can be variationally determined (coefficients C optimized, orbitals fixed) by solving the matrix eigenvalue equation resulting from differentiating the standard Hamiltonian energy expression with respect to the elements of the CI coefficient vector C and setting the result to zero:
where H is a semi-empirical Hamiltonian matrix over microstates (Hpq = <ψp|H|ψq>) and V is the overlap matrix over microstates (Vpq = <ψp|ψq>).
Members of the set of CI wavefunctions satisfying Equation 10.2 are called "CI eigenstates" and are labeled here by their root number in order of increasing energy as Ψ[R], starting with R = 1 for the lowest energy CI eigenstate Ψ[1].
A microstate ψ with Nα alpha electrons and Nβ beta electrons is an eigenfunction of the operator Ŝz (the z-component of the total electron spin angular momentum) with eigenvalue Sz:
For an N-electron system, the set {Sz} of possible values for Sz is:
A microstate ψ is not an eigenfunction of the operator Ŝ2 (the square of the total electron spin angular momentum) unless it has a closed shell configuration (all MOs doubly occupied or empty) or a high-spin open-shell configuration (all singly-occupied MOs have parallel spin):
A "spin adapted microstate" η[S,Sz] is a linear combination of microstates that is defined to be an eigenfunction of both Ŝz and Ŝ2:
For example, a microstate with two singly-occupied MOs with opposite spin (Sz = 0) is not an eigenfunction of Ŝ2, but the combination of this microstate with the corresponding spin-flipped microstate is an eigenfunction of Ŝ2, with quantum number S = 0.
For an N-electron system, the set of possible values for S are:
For a given value of S, the set of possible values of Sz are:
The "spin multiplicity" SM corresponding to S is given by:
The spin multiplicity indicates the number of possible values for Sz, and therefore the degeneracy of a spin adapted microstate with total spin quantum number S. For "singlets" (SM = 1, S = 0), there is only one possibly value for Sz: Sz = 0. For "doublets" (SM = 2, S = 1/2), there are two possible values for Sz: Sz = -1/2 and Sz = 1/2. For "triplets" (SM = 3, S = 1), there are three possible values for Sz: Sz = -1, Sz = 0 and Sz = 1.
Since exact eigenstates of the non-relativistic Hamiltonian (modeled by the semi-empirical Hamiltonian H) are pure spin states, i.e., eigenfunctions of both Ŝz and Ŝ2, it useful to constrain the CI eigenstates to be as well. This can be achieved if, instead of using just the "raw" microstates ψ as the N-electron basis prior to solving Equation 10.2, the N-electron basis is defined in terms of spin adapted microstates η.
In terms of a set of spin adapted microstates η[S,Sz] with spin quantum numbers S and Sz, a corresponding pure spin state CI eigenstate Ψ[R,S,Sz]CI is given by:
In AMPAC, the CI eigenstates generated are always expanded according to Equation 10.13 and so they are pure spin states. In addition, for efficiency only one member of a degenerate set of spin adapted microstates is ever used. By default, this is the one with the smallest non-negative value of Sz (0 for even-electron systems, 1/2 for odd-electron systems), but this is modifiable using the keywords SZ=n or MICROS=n. Thus, the CI matrix equations to be solved are:
where H[R,S,Sz]pq = <ηp[S,Sz]|H|ηq[S,Sz]> and Vpq = <ηp[S,Sz]|ηq[S,Sz]>.
Combining equations 6.7 and 6.13, Ψ[R,S,Sz]CI can be expressed directly in terms of the microstates ψm[Sz] with coefficients Dm[R,S,Sz] as:
In the AMPAC output files, it is always the microstate coefficients Dm[R,S,Sz] which are printed.
While there are many CI eigenstates which can be calculated (the number can be specified using the keyword CISTATE=n), AMPAC considers one of them to be the "primary CI eigenstate" whose energy hypersurface will be followed during geometry optimizations and which will be used as the reference for all property calculations. The other n - 1 CI eigenstates requested by CISTATE=n are considered "secondary" CI eigenstates, for which some properties are calculated and printed, typically at one or more optimized geometries of the primary eigenstate, so their transition properties are non-adiabiatic. By default, the primary CI eigenstate is the ground state, independent of spin multiplicity, and the secondary eigenstates are all excited states. To specify a different primary CI eigenstate, use one of the spin multiplicity keywords SINGLET, DOUBLET, TRIPLET, etc. and / or ROOT=n, where n = 1 refers to the ground state. For example, to use the secondlowest energy triplet CI eigenstate ("T2") as the primary one, specify TRIPLET and ROOT=2. To use the secondlowest energy CI eigenstate of any spin multiplicity as the primary one, specify ROOT=2 without a spin multiplicity keyword.
For the CI eigenstate Ψ[R,S,Sz], the total electron density function ρ[R,S,Sz](r) is expressed in terms of the 2M occupied and virtual SOs χ of ΨRef by:
where γmi is the occupancy (0 or 1) of the ith SO for the mth microstate, sz,i = 1/2 for alpha SOs and -1/2 for beta SOs. In terms of the corresponding M MOs, ρ[R,S,Sz](r) is given by:
where γa,mi is the occupancy (0 or 1) of the alpha SO of the ith MO for the mth microstate and PMO[R,S,Sz] is the total one-electron density matrix in the MO basis, with alpha and beta contributions PMO,α[R,S,Sz] and PMO,β[R,S,Sz], respectively. In terms of L AOs, ρ[R,S,Sz](r) is given by:
where PAO[R,S,Sz] is the total one-electron density matrix in the AO basis, with alpha and beta contributions PAO,α[R,S,Sz] and PAO,β[R,S,Sz], respectively.
In AMPAC, when the keyword CIDIP is specified, the dipole moment and Mulliken atomic charges are calculated for both the primary and secondary CI eigenstates from the corresponding density matrices PAO,β[R,S,Sz]. In general, other one-electron properties which are also available without CI, such as ESP charges, are calculated in CI calculations from PAO,β[R,S,Sz], but only for the primary CI eigenstate.
The "electron spin density" ρs[R,S,Sz](r) corresponding to ρ[R,S,Sz](r) is simply the alpha electron density ρα[R,S,Sz](r) minus the beta electron density ρβ[R,S,Sz](r), which, along with the corresponding spin density matrices PMO,S[R,S,Sz] and PAO,S[R,S,Sz] is given by:
In AMPAC, when the ESR keyword is specified, the spin density matrices PMO,S[R,S,Sz] and PAO,S[R,S,Sz] are printed for the primary CI eigenstate along with the net Mulliken atomic electron spins for both primary and secondary CI eigenstates. The net Mulliken electron spin for the Ath atom, σA, is calculated like the corresponding Mulliken atomic electron population except that PAO,S[R,S,Sz] is used instead of PAO[R,S,Sz]:
The transition dipole moment between CI eigenstates Ψ[R,S,Sz] and Ψ[n,S,Sz] is an important result:
For example, contributions from all available transition dipole moments appears in the "sum-over-states" (SOS) expression for the dynamic polarizability tensor αSR(ω), given by Equation 10.24. Individual transition dipole moments are also of interest because they yield information about the UV / visible spectrum of a molecule. The oscillator strength between states Ψ[R,S,Sz] and Ψ[n,S,Sz] is proportional to the absorptivity of light at a wavelength λ[R->n,S,Sz]:
where K is a constant. By default, AMPAC writes the transition dipole moments μ[R->n,S,Sz], transition wavelengths λ[R->n,S,Sz] and oscillator strengths f[R->n,S,Sz] between the primary eigenstate Ψ[R,S,Sz] and all of the secondary CI eigenstates Ψ[n,S,Sz]. In AMPAC, the number of CI eigenstates to calculate, including the primary CI eigenstate, can be specified using the CISTATE=n keyword (some of these will have a different total spin quantum number S than the primary eigenstate and so their corresponding transition dipole moments vanish).
The "sum-over-states" (SOS) expression for the dynamic polarizability tensor α[R,S,Sz](ω) for the CI eigenstate Ψ[R,S,Sz] is given by:
where ω is the external electric field frequency (in energy units) and the sum is over all possible CI eigenstates different from the primary eigenstate, but having the same S and Sz quantum numbers. In AMPAC, αSR(ω) will be calculated and written to the AMPAC output file when the keywords DYNPOL or DYNPOL=n.nnnn are specified. Note that the keyword CISTATE=n has no influence on the calculation of dynamic polarizabilities, and vice versa, but the number of possible CI eigenstates (determined by the active space and hence the number of final microstates) does.
The set of occupied and virtual MOs whose corresponding SOs are allowed to exchange electrons in ΨRef to form new microstates ψ are called the CI-active MOs or the "active space". The choice of active space is one of the most crucial, and sometimes difficult, steps in a CI calculation, both computationally and in terms of physical results. Given this importance, the CI-active MOs are usually specified along with the keywords which invoke CI, possibly together with the RECLAS(n.m) keyword and its associated MO permutation data. For example, C.I.(5,8) means "do a CAS-CI using MOs 5,6,7 and 8 as the CI-active MOs". It is essential that all or none of the members of a degenerate set of MOs be included in the active space. By default, AMPAC will abort if this is not the case. The keywords, CIGAP=n.nnnn and CI-OK can be used to alter the definition of MO degeneracy and to allow the active space to contain an incomplete set of degenerate MOs. By default, all of the MO energies are printed to the AMPAC output file. The keywords VECTORS and ALLVEC can be used to print both the MO energies and AO coefficients to the AMPAC output file for inspection. This information is also present in an AMPAC visualization file so that MOs can be visualized with AMPAC's GUI. It is important to know the order in which the SCF MOs occur and their corresponding labeling. For a system with M MOs, the MOs are ordered from 1 to M by increasing occupancy, i.e., first doubly-occupied MOs, then partially occupied MOs and finally unoccupied (virtual) MOs. This order usually coincides with increasing MO energy for the entire list from 1 to M, but within each subset of the same occupancy the order always coincides with increasing energy.
For RHF open-shell calculations, the SCF calculations in AMPAC are done using the "half-electron" method. In this method, the usual "spin-less" closed-shell RHF SCF formalism is used to calculate ΨSCF, except that instead of N / 2 doubly occupied spatial orbitals there are assumed to be N / 2 - n (N even) or N / 2 + 1 - n (N odd) doubly occupied orbitals and m orbitals with an occupancy of n / m. These fractionally occupied orbitals may be thought of as being occupied by two "half-electrons" of opposite spin and with a charge of n / 2m. This leads to an energy expression which is similar to Roothan's multiconfiguration open-shell SCF energy expression after spurious coulomb and exchange energies arising from the interaction between "half-electrons" are subtracted out. In AMPAC, however, the energy calculated using the "half-electron" method is never used, since it is non-variational, but the corresponding set of SCF orbitals are, either in a "minimal" CAS-CI calculation involving all of the partially occupied MOs of ΨSCF as the active space if CI is not otherwise invoked, or more generally in any specified type of CI calculation. While the fractionally occupied SOs of ΨSCF determine the active space of corresponding MOs, the reference wavefunction ΨRef for open-shell RHF is not ΨSCF but rather a determinant obtained by filling the SOs of ΨSCF in the standard way using the Aufbau principle. It is important to note that, in general, the number of open-shell electrons to assume for the SCF should be specified explicitly using one of the keywords OPEN(n.m), BIRADICAL or EXCITED, otherwise AMPAC will assume the minimum number of open-shell electrons (0 for even-electron systems and 1 for odd-electron systems) for the SCF. The spin-multiplicity keywords (e.g., SINGLET, DOUBLET, TRIPLET, etc.) are not used in RHF until the CI portion of the calculation. Thus, for the oxygen molecule, OPEN(2,2) should be specified even if TRIPLET is also specified.
Given ΨRef and a corresponding active space, a definition of which microstates to generate and potentially use for the expansion of the CI eigenstates is necessary. In the "Complete Active Space" method (CAS-CI), specified by C.I.=n or C.I.(n,m) and the default when CI is only implied by OPEN(n,m), all possible microstates which can be generated by permutations of the electrons among the SOs within the active space are potentially used. In the "CI Singles" method (S-CI), specified by SC.I.=n or SC.I.(n.m), all possible singly-excited microstates ψia are potentially used. In the "CI Singles and Doubles" method (SD-CI), specified by SDC.I.=n or SDC.I.(n,m), all possible singly-excited microstates ψia and doubly-excited microstates ψia,jb are potentially used. In the "CI Singles, Doubles and Triples" method (SDT-CI), specified by SDTC.I.=n or SDTC.I.(n,m),all possible singly-excited microstates ψia, doubly-excited microstates ψia,jb and triply-excited microstates ψia,jb,kc are potentially used. The initial set of microstates is referred to here as {I}MS.
The size of {I}MS grows very rapidly (combinatorially) as the size of the active space increases, especially when CAS-CI is used. (For a CAS-CI involving 10 electrons and 10 CI-active MOs, the number of possible microstates is over 60000, after spin degeneracies are excluded.) In some cases, all of {I}MS should be used, if possible. If this is not the case, whether due to resource limitations and / or to avoid "over-correlating" the already partially correlated, semi-empirically calculated ground state energy, then some means of efficiently selecting the most important "final" set of microstates, referred to here as {F}MS, from {I}MS is necessary. Typically, only a relatively small "target" set of the possible CI eigenstates, {R}ES, are of interest. For example, {R}ES might be composed of the singlet ground CI eigenstate and the first excited singlet and triplet CI eigenstates. {R}ES can usually be characterized in terms of relatively large contributions from a small subset of "germ" microstates {G}MS = {G0, G1, G2, ?}MS, where G0 ≡ ΨRef roughly corresponds to the ground CI eigenstate R0, G1 to a first excited CI eigenstate R1, etc. While, much of the information relevant to {R}ES is included in {G}MS, the CI eigenstates of {R}ES constructed from {G}MS alone would generally have two significant deficiencies. First, there is generally a lack of specific correlation within the set {G}MS. Second, the excited members {G}MS are lacking in "repolarization" because the SCF orbitals from which {G}MS is generated are obtained from a ground state wavefunction optimization. The objective of the microstate selection procedure used in AMPAC to produce {F}MS is to extract from the enormous list of initial microstates in {I}MS and not in {G}MS, the ones which should contribute most to specific correlation and repolarization. This microstate selection consists of four major steps:
From the initial microstate space {I}MS, keep those J1 (≈ 10 × J4, J4 defined below) microstates ψ with the lowest MøllerPlesset zeroorder energy E0MP[ψ] (sum of occupied SO energies):
where the sums over i and j are over all alpha and beta SOs, respectively, while λ and ε represent SO occupancies and energies, respectively.
From the J1 microstates of step I, choose the J2 (default 100) microstates ψ with the lowest EpsteinNesbet (EN) energy EEN[ψ] (semiempirical Hamiltonian expectation value).
This set of J2 microstates is the "germ" set {G}MS referred to above.
From {G}MS of step II, determine the J3 (default 30) eigenvectors of the corresponding CI matrix
From the J1 - J2 "non-germ" microstates ψ which are in {I}MS but not {G}MS, choose the J4 (default 1200) - J2 microstates which make the largest contribution to the following quantity W[ψ]:
At each stage of this microstate selection procedure, the sets of microstates selected are required to preserve spatial degeneracy, i.e., all members of a degenerate set of microstates are kept if there is space available in the target list, or not kept if there is not space available in the target list. This is achieved by simple inspection of the Møller-Plesset zero-order energies, using a degeneracy threshold of 1.0 × 10-4 eV, which is adjustable by the keyword CIGAP=n.n Of course, this procedure will not cover the case of an active space containing only a partial set of degenerate MOs. It is important to remember that either all or none of the members of a degenerate set of MOs should be included in the active space.
In AMPAC, the above microstate selection procedure can be partially customized by specifying the parameters J2, J3 and J4 using the keywords CIMAX=J4, PERTU=J2 and PERTU(J2,J3).
Specifying any of the following keywords together with a keyword which either explicitly or implicitly invokes CI is an error, and will result in AMPAC aborting the job.
| COSMO | CI is not available with solvation models. |
| SM5.2 | CI is not available with solvation models. |
| SM5.2R | CI is not available with solvation models. |
| SM5C | CI is not available with solvation models. |
| SM5CR | CI is not available with solvation models. |
| SCFLOCAL | Not available for methods using analytical CI gradients. |
| UHF | CI requires RHF. |
| RESTART | CI calculations cannot be reliably restarted. |
| BIRADICAL | System has two unpaired electrons. |
| C.I. | Include n orbitals around the HOMO in the CI manifold. |
| CI-OK | Override degeneracy check. |
| CIDIP | Calculate charges and dipole moments for CI eigenstates. |
| CIGAP | Specify energy gap used to determine microstate degeneracy. |
| CIMAX | Specify the maximum number of microstates. |
| CIOUT | Write details about the CI eigenstates to file. |
| CISTATE | Specify the number of final CI eigenstates to be calculated and printed. |
| DAVDBG | Write details about the CI matrix diagonalization to file. |
| DECET | RHF decet state required. |
| DOUBLET | RHF doublet state required |
| DYNPOL | Outputs data for dynamic polarizability calculations. |
| ESR | Unpaired spin density on atoms will be calculated. |
| EXCITED | First excited singlet state will be optimized. |
| FILL | Require use of defined set of prototype MOs. |
| INCI | Read final microstates from an ASCII file. |
| JKPRINT | All unique two electron integrals over CI-active MOs written to output file. |
| MATCI | Energies and AO coefficients of CI-active MOs printed to output file. |
| MECI | Print information about CI microstates and transitions. |
| MICROS | Generates only microstates with spin = n. |
| MSCHARG | Maximum charge for generated microstates |
| n-ET | Constrains the spin multiplicity of the primary CI eigenstate to be n. |
| NONET | RHF nonet state required |
| OCTET | RHF octet state required |
| OPEN | Configuration Interaction |
| PROTO | Define prototype MOs. |
| PERTU | Override the default perturbative selection of microstates. |
| QUARTET | RHF quartet state required |
| QUINTET | RHF quintet state required |
| RECLAS | Reorder MOs. |
| RIGIDCI | Propagate initial selection of microstates throughout a geometry optimization. |
| ROOT | Specify spin state to follow. |
| SCFCI | Define partial occupancy of the "virtual" CI-active MOs for a "half-electron" RHF SCF calculation. |
| SC.I. | Specify CI-active MOs in a S-CI calculation. |
| SDC.I. | Specify CI-active MOs in a SD-CI calculation. |
| SDTC.I. | Specify CI-active MOs in a SDT-CI calculation. |
| SEPS | Specify energy gap used to determine eigenstate degeneracy. |
| SINGLET | RHF singlet state required |
| SEPTET | RHF septet state required |
| SEXTET | RHF sextet state required |
| SZ | Specify value of Sz. |
| TRIPLET | Triplet state required |
| VALIDCI | Indicate that the microstates to be read in are fully consistent. |
AMPAC was run on the following AMPAC input file:
am1 c.i.=2 cistate=3 cimax=100 singlet t=auto truste lforce bonds D2h Ethylene, AM1/CI, HOMO LUMO Active, S0 (Singlet Ground State) Opt+LowFreq, Active MOs and Bonds, Calc 3 Lowest States C 0.000000 0 0.000000 0 0.000000 0 0 0 0 C 1.325916 1 0.000000 0 0.000000 0 1 0 0 H 1.098266 1 122.715971 1 0.000000 0 1 2 0 H 1.098266 1 122.715971 1 -180.000000 1 1 2 3 H 1.098266 1 122.715971 1 0.000000 1 2 1 4 H 1.098266 1 122.715971 1 180.000000 1 2 1 4 0 0.000000 0 0.000000 0 0.000000 0 0 0 0
Timestamp: 2004-02-16-16-44-34-0000019206-hpux
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AM1 CALCULATION RESULTS
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* AMPAC Version 8.13
* Presented by:
*
* Semichem, Inc.
* PO Box 1649
* Shawnee KS 66222
* (913)268-3271
* (913)268-3445 (fax)
*
* TRUSTE - MINIMISE ENERGY USING TRUST REGION
* LFORCE - LOWEST IR FREQUENCIES CALCULATION SPECIFIED
* T=AUTO - AUTOMATIC DETERMINATION OF ALLOWED TIME
* C.I.=N - 2 M.O.S TO BE USED IN C.I.
* CIMAX= 100 - ALLOWED SIZE FOR CI MATRIX
* CISTATE= 3 - EIGENSTATES CALCULATED IN CI
* BONDS - FINAL BOND-ORDER MATRIX TO BE PRINTED
* SINGLET - IS THE REQUIRED SPIN MULTIPLICITY
* AM1 - THE AM1 HAMILTONIAN TO BE USED
*******************************************************************************
AM1 C.I.=2 CISTATE=3 CIMAX=100 SINGLET T=AUTO TRUSTE LFORCE BONDS
D2h Ethylene, AM1/CI, HOMO LUMO Active, S0 (Singlet Ground State)
Opt+LowFreq, Active MOs and Bonds, Calc 3 Lowest States
ATOM CHEMICAL BOND LENGTH BOND ANGLE TWIST ANGLE
NUMBER SYMBOL (ANGSTROMS) (DEGREES) (DEGREES)
(I) NA:I NB:NA:I NC:NB:NA:I NA NB NC
1 C
2 C 1.32592 * 1
3 H 1.09827 * 122.71597 * 1 2
4 H 1.09827 * 122.71597 * -180.00000 * 1 2 3
5 H 1.09827 * 122.71597 * 0.00000 * 2 1 4
6 H 1.09827 * 122.71597 * 180.00000 * 2 1 4
MOLECULAR POINT GROUP SYMMETRY CRITERIA
D2H 0.10000000
SINGLET STATE CALCULATION
RHF CALCULATION, No. OF DOUBLY OCCUPIED LEVELS = 6
** REFERENCES TO PARAMETERS **
H (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)
C (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)
CARTESIAN COORDINATES
NO. ATOM X Y Z
1 6 0.0000 0.0000 0.0000
2 6 1.3259 0.0000 0.0000
3 1 -0.5936 0.9240 0.0000
4 1 -0.5936 -0.9240 0.0000
5 1 1.9195 -0.9240 0.0000
6 1 1.9195 0.9240 -0.0000
STANDARD DEVIATION ON ENERGY (KCAL) 0.00000055519
STANDARD DEVIATION ON GRADIENT (KCAL/A,RD,RD) 0.00008233 0.00007642 0.00008301
LOWEST IR FREQUENCIES CALCULATION (MARCH 1999)
HEAT OF FORMATION= 8.261415 kcal/mole
RMS GRADIENT NORM= 0.020640 kcal/mole/A
HESSIAN SPANNED BY 12 INTERNAL COORDINATES.
1 LOWEST EIGENVALUES OF THE HESSIAN HAVE BEEN ACCURATELY CALCULATED.
NON ZERO EIGENVALUES, (STD DEV) AND ASSOCIATED EIGENVECTORS: (Angstroms or radians)
1.51D+01 -0.000 0.000 0.000 0.000 0.000 0.703 -0.000 -0.000 0.599 -0.000
(1.44D-03) -0.000 0.384
NOTE: WAVE NUMBERS ARE BIASED WITH RESPECT TO EXACT VALUES,
BUT SIGNS ARE ASCERTAINED (UNLESS A ERROR BAR TOO LARGE).
VIBRATIONAL FREQUENCIES AND ERRORS (CM-1),
REDUCED FORCE CONSTANTS (MILLIDYNE/ANGSTROMS),
DIPOLE DERIVATIVES (DEBYE/ANGSTROMS),
AND NORMAL MODES (CARTESIAN COORDINATES):
FREQ : -0.000 -0.000 -0.000 0.000 0.000 0.000 965.292
ERROR : 0.000 0.000 0.000 0.000 0.000 0.000 0.137
F-CST : -0.00000 -0.00000 -0.00000 0.00000 0.00000 0.00000 0.27450
DIP(X): 0.000 -0.000 -0.000 0.000 0.000 0.000 -0.000
DIP(Y): 0.000 -0.000 -0.000 -0.000 0.000 -0.000 -0.000
DIP(Z): 0.000 -0.000 0.000 -0.000 0.000 -0.000 -0.000
DIP TOT 0.000 0.000 0.000 0.000 0.000 0.000 0.000
1C (x) 0.0158 -0.0251 -0.0272 0.0122 0.0076 -0.1839 -0.0000
1C (y) -0.1866 -0.1327 -0.0477 0.0443 -0.0249 0.0110 -0.0000
1C (z) -0.1466 0.1494 0.0917 -0.0760 -0.0319 -0.0528 0.1244
2C (x) 0.0158 -0.0251 -0.0272 0.0122 0.0076 -0.1839 -0.0000
2C (y) -0.0596 0.0165 0.0716 0.2152 0.0465 -0.0017 0.0000
2C (z) -0.0681 0.0793 -0.0934 -0.0375 0.2018 0.0030 -0.1244
3H (x) -0.0720 -0.1281 -0.1095 -0.1058 -0.0417 -0.1751 0.0000
3H (y) -0.2424 -0.1982 -0.1001 -0.0307 -0.0562 0.0166 -0.0000
3H (z) -0.1922 0.4130 -0.1736 0.0511 -0.3636 -0.0588 -0.3948
4H (x) 0.1035 0.0779 0.0552 0.1302 0.0569 -0.1927 0.0000
4H (y) -0.2424 -0.1982 -0.1001 -0.0307 -0.0562 0.0166 0.0000
4H (z) -0.1699 -0.0527 0.5196 -0.2370 0.0947 -0.0959 -0.3948
5H (x) 0.1035 0.0779 0.0552 0.1302 0.0569 -0.1927 0.0000
5H (y) -0.0038 0.0819 0.1239 0.2902 0.0779 -0.0073 -0.0000
5H (z) -0.0225 -0.1843 0.1718 -0.1646 0.5336 0.0090 0.3948
6H (x) -0.0720 -0.1281 -0.1095 -0.1058 -0.0417 -0.1751 0.0000
6H (y) -0.0038 0.0819 0.1239 0.2902 0.0779 -0.0073 0.0000
6H (z) -0.0447 0.2813 -0.5213 0.1235 0.0753 0.0461 0.3948
AM1 C.I.=2 CISTATE=3 CIMAX=100 SINGLET T=AUTO TRUSTE LFORCE BONDS
D2h Ethylene, AM1/CI, HOMO LUMO Active, S0 (Singlet Ground State)
Opt+LowFreq, Active MOs and Bonds, Calc 3 Lowest States
GEOMETRY OPTIMISED : ENERGY MINIMISED
SCF FIELD WAS ACHIEVED
AM1 CALCULATION
VERSION 8.13
Feb-16-2004
FINAL HEAT OF FORMATION = 8.261415 kcal (CI SINGLET No 1) 
= 34.574020 kJ
ELECTRONIC ENERGY = -736.450177 eV
CORE-CORE REPULSION = 425.733180 eV
TOTAL ENERGY = -310.716996 eV
GRADIENT NORM = 0.071496
RMS GRADIENT NORM = 0.020639
UNSTABLE MODE(S) = 0 ( ACCURATE )
MOLECULAR POINT GROUP = D2H 0.100000
NO. OF FILLED LEVELS = 6 (OCC = 2)
MOLECULAR WEIGHT = 28.054
SCF + CI CALCULATIONS = 16
COMPUTATION TIME = 0.05 seconds
ATOM CHEMICAL BOND LENGTH BOND ANGLE TWIST ANGLE 
NUMBER SYMBOL (ANGSTROMS) (DEGREES) (DEGREES)
(I) NA:I NB:NA:I NC:NB:NA:I NA NB NC
1 C
2 C 1.34092 * 1
3 H 1.09708 * 122.43943 * 1 2
4 H 1.09708 * 122.43943 * -180.00000 * 1 2 3
5 H 1.09708 * 122.43943 * 0.00000 * 2 1 4
6 H 1.09708 * 122.43943 * 180.00000 * 2 1 4
MOLECULAR POINT GROUP SYMMETRY CRITERIA
D2H 0.10000000
RHF EIGENVALUES
-32.96035 -21.92659 -15.75830 -14.25910 -11.89512 -10.45383 1.37797 4.05778
4.39209 5.06914 5.56126 5.74277
CONFIGURATION INTERACTION CALCULATION 
4 MICRO-STATES GENERATED BY CAS-CI VS 44 ROOM AVAILABLE.
4 MICRO-STATES FINALLY KEPT.
CI-ACTIVE MOLECULAR ORBITALS:
ROOT NO. 1 2 
-10.454 1.378
s C 1 -0.0000 0.0000
Px C 1 0.0000 -0.0000
Py C 1 -0.0000 -0.0000
Pz C 1 -0.7071 0.7071
s C 2 0.0000 -0.0000
Px C 2 -0.0000 0.0000
Py C 2 0.0000 0.0000
Pz C 2 -0.7071 -0.7071
s H 3 -0.0000 -0.0000
s H 4 0.0000 -0.0000
s H 5 -0.0000 0.0000
s H 6 0.0000 -0.0000
DETAILED COUNT OF THE 4 CALCULATED LOWEST EIGENSTATES: 
SINGLET TRIPLET
CSF 3 1
STATES 3 1
THE EIGENSTATE SELECTED IS No 1 (SINGLET)
ROW: MAIN MICRO-STATES OVER THE 4 SELECTED IN C.I.
COLUMN: EIGENSTATES FROM 1 TO 3
MO: 00 1:SINGLET 2:TRIPLET 3:SINGLET 
: 67 eV: 0.0000 2.9456 6.6045
1 20 96% 0% 0%
( 0.9807) ( 0.0000) ( 0.0000)
2 +- 0% 50% 50%
( 0.0000) ( 0.7071) (-0.7071)
3 -+ 0% 50% 50%
(-0.0000) ( 0.7071) ( 0.7071)
4 02 4% 0% 0%
(-0.1953) ( 0.0000) ( 0.0000)
TRANSITION DIPOLE (A.U) AND OSC. STRENGTHS FROM STATE 1 (SINGLET) TO OTHERS 
STATE eV nm X Y Z STRENGTH
2 2.946 420.9 FORBIDDEN TO TRIPLET
3 6.605 187.7 1.407 0.000 -0.000 0.3205
SUM OF STRENGTHS: 0.9615 0.0000 0.0000
NET ATOMIC CHARGES AND DIPOLE CONTRIBUTIONS 
ATOM NO. TYPE CHARGE ATOM ELECTRON DENSITY
1 C -0.2193 4.2193
2 C -0.2193 4.2193
3 H 0.1097 0.8903
4 H 0.1097 0.8903
5 H 0.1097 0.8903
6 H 0.1097 0.8903
DIPOLE (DEBYE) X Y Z TOTAL
POINT-CHG. 0.000 -0.000 -0.000 0.000
HYBRID 0.000 0.000 -0.000 0.000
SUM 0.000 -0.000 -0.000 0.000
CARTESIAN COORDINATES 
NO. ATOM X Y Z
1 C 0.0000 0.0000 0.0000
2 C 1.3409 0.0000 0.0000
3 H -0.5885 0.9259 0.0000
4 H -0.5885 -0.9259 -0.0000
5 H 1.9294 -0.9259 -0.0000
6 H 1.9294 0.9259 0.0000
ATOMIC ORBITAL ELECTRON POPULATIONS 
1.24839 0.95422 1.01669 1.00000 1.24839 0.95422 1.01669 1.00000
0.89035 0.89035 0.89035 0.89035
BOND ORDERS AND VALENCIES 
C 1 C 2 H 3 H 4 H 5 H 6
------------------------------------------------------------------------------
C 1 3.928231
C 2 1.852496 3.928231
H 3 0.957752 0.006722 0.987976
H 4 0.957752 0.006722 0.008643 0.987976
H 5 0.006722 0.957752 0.013244 0.001615 0.987976
H 6 0.006722 0.957752 0.001615 0.013244 0.008643 0.987976
FULL COMPUTATION TIME : 0.05 SECONDS
Process Info: 0.1u 0.2s 0:00 25%
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The primary CI eigenstate is, as requested, the ground state singlet, S0. For reference, the AM1/SCF and experimental heats of formation are 16.5 kcal/mol and 12.5 kcal/mol, respectively. There is a significant decrease in energy in going from AM1/SCF to even this "minimal" AM1/CAS-CI. Since AM1 (and all of the semi-empirical models in AMPAC) was parameterized against experiment at the SCF level, absolute heats of formation at the corresponding CI level are generally too low, especially at higher levels of CI. |
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This Z-matrix shows the AM1 optimized geometry of Ethylene in the primary CI eigenstate. All of the results in AMPAC output file, including those for the secondary CI eigenstates, are calculated at this geometry. The transition energies between the primary and secondary CI eigenstates are thus "vertical" transition energies. |
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As requested, the number of CI-active MOs was 2, the HOMO and LUMO. There are six possible microstates, given below in terms of the SO occupancies of the HOMO and LUMO, where + means the alpha SO is occupied, - means the beta SO is occupied and 0 means the corresponding SO is unoccupied: Table 10.1. Possible microstates with 2 CI-active MOs
This "minimal" CAS-CI for a singlet state uses the first four 4 microstates, which have Sz = 0. The 5th and 6th microstates, with Sz = 1 and -1, are degenerate with the linear combination of the 3rd and 4th microstates and so are not used, even for the triplet CI eigenstates (they cannot be used for the singlet states, of course). |
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The AO coefficients of the two CI-active MOs show that the first (the HOMO) is a π MO while the second (the LUMO) is a corresponding π* MO. |
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The 3 lowest secondary CI eigenstates were calculated in addition to the primary one S0. One of the secondary CI eigenstates is a triplet, the others being singlets. The "CSF" row in the table refers to the number of spin-adapted configurations used in the expansion of the CI eigenstates. |
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This table gives the contribution - as both a percentage and as a normalized coefficient - of each microstate to each of the 3 requested CI eigenstates, whose energies are given (in eV) relative to the primary eigenstate. The SO occupancies of the CI-active MOs for each microstate are also given. Thus, the first excited state, "2:TRIPLET" (T1), has an energy of -307.7714 (= -310.716996 + 2.9456) eV and it is an equal combination (50% / 50%) of the the second and third microstates. In the second microstate, the MOs 6 (HOMO) and 7 (LUMO) are both singly occupied ("+-" means that only the alpha SO of the HOMO is occupied while only the beta SO of the LUMO is occupied). The third microstate is the "spinflipped" complement to the second, with the beta SO of the HOMO being occupied and the alpha SO of the LUMO being occupied. The primary CI eigenstate, "1:SINGLET" (ground state S0), is nearly identical to the first microstate, which is the SCF reference determinant ("20" means 2 electrons in the HOM0 and 0 electrons in the LUMO). The fourth microstate, in which the LUMO is doubly occupied, makes only a 4% contribution to the primary CI eigenstate and no contribution to the others. Note that the Sz eigenvalue is 0 for all microstates. |
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This table gives the transition energies, transition wavelengths, transition dipoles and oscillator strengths between the primary CI eigenstate and the two secondary CI eigenstates. Since the primary CI eigenstate is a singlet, the transition dipole and corresponding oscillator strength for the triplet CI eigenstate is identically zero. The first excited singlet state (S1) has a significant transition dipole parallel to the C-C bond and the corresponding oscillator strength of 0.3025 indicates there should be a significant absorption intensity around 187 nm for gas phase ethylene. |
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The atomic Mulliken charges and dipole moments given in this section are for the primary CI eigenstate. Values for the secondary eigenstates can be calculated with the AMPAC keyword CIDIP. |
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This table gives the atoms' Cartesian coordinates for the geometry optimized in the primary CI eigenstate. |
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These tables of AO electron populations, bond orders and valencies are calculated from the first-order density matrix of the primary CI eigenstate. |
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| Electrostatic Potential |  |
Polarizability Methods |
Copyright © 1994, 1997, 2004, Semichem, Inc. All rights reserved. | ||