On the calculation of energies of excited states

We reformulate a scheme for calculation of the energies of excited states which, unlike the variation method, does not require that the trial function be orthogonalized to the wavefunctions for lower excited states. The possibility of obtaining wavefunctions as well as energies is discussed, and an example of the scheme's application is made to the harmonic oscillator.

---

Zusammenfassung Es wird eine neue Form für eine Methode zur Berechnung von angeregten Zuständen gegeben. Die Probefunktion braucht nicht wie bei der Variationsmethode orthogonal zu den Wellenfunktionen der tiefer liegenden Zuständen zu sein. Es wird die Möglichkeit diskutiert, Wellenfunktionen ebenso wie Energien zu erhalten. Das Schema wird auf den harmonischen Oszillator angewandt.

---

Resume On donne une nouvelle forme pour une methode de calcul des niveaux excités. La methode ne demande pas l'orthogonalisation de la fonction d'essai par rapport aux fonctions d'onde correspondant aux niveaux inférieurs, ce qui montre un avantage auprès de la methode variationnelle. La possibilité d'obtenir les fonctions d'onde elles-mêmes est discutée, et on donne un exemple de l'application de cette nouvelle methode.

     

Pseudomolecular atoms: geometry of two-electron intrashell excited states

Supermultiplet patterns indicating pronounced collective motion of electrons are prominent among intrashell excited states of helium or alkaline earth atoms. Many aspects of these patterns have previously been discussed in terms of an empirical rovibrator model, e-core-e, analogous to alinear triatomic molecule. However, this has appeared incompatible with a dimensional scaling treatment that predicts pseudomolecular features and relates properties of some excited states to the ground state, but corresponds to abent equilibrium geometry. We examine both the full two-electron Hamiltonian, including angular momentum, and a prototype model for angular correlation which restricts the two particles to the surface of a sphere. By virtue of the boundary conditions imposed by the Jacobian volume element, we find that alinear equilibrium geometry is not allowed. If the Hamiltonian is transformed to reduce the Jacobian to unity, an infinite barrier appears in the effective potential when the electrons are 180° apart and equidistant from the nucleus. We find the supermultiplet patterns, including some “antimolecular” features, are consistent with a floppy butbent geometry, anasymmetric rotor model. The most probable interelectron angle ? _m varies markedly with the principal quantum number and also with the space-quantization of the angular momentum with respect to body-fixed axes.     

An SCF technique for excited states

A method for deriving HF-SCF wave functions for excited states is presented here. All the active orbital transformations that are compatible with the orthogonality requirements are performed without unnecessary restrictions on the variational space and within a direct minimization approach. The method has been tested with an application on the first excited-singlet state of Be.     

Converging SCF calculations on excited states

A practical scheme for the calculation of excited states of the same symmetry as a given reference state is outlined in the context of the Hartree-Fock method. In order to prevent the excited state from “collapsing” into a lower-lying state, the prediagonalized Fock matrix is diagonalized in a restricted subspace, deleting the component associated with the orbital which participates in the excitation. Computationally, the deletion is accomplished by means of a “big shift” of the associated diagonal element of the prediagonalized Fock matrix. The resulting wave function will not be fully relaxed, but can be shown to be orthogonal to the reference state. The method has been implemented in a molecular UHF program. Applications to the 4σ hole state of CO and to an excited state of the CuCl ion are reported.     

Perturbation and SCF calculations for excited states

One can treat the linear variation calculation by the perturbation expansion; on the other hand, one can also relate the perturbed wave function for the mth state to the mth eigenvector and justify the upper bound nature of the perturbation calculations for excited states. One can also associate a special type linear variation wave function to the SCF calculation. The upper bound nature of the excited-state SCF calculation can then be accessed by the help of this special wave function.     

On the calculation of the excited states of small molecules

The excited states of CO, H₂O and NH₃ have been calculated by the singly excited configuration interaction method using two large basis sets, one of which contained diffuse functions in order to describe Rydberg states. Results are found to be in good agreement with experiment for both excitation energies and oscillator strengths for the transitions from the ground state.     

Singlet excited states of silicon-containing anions relevant to interstellar chemistry

As the number of anions detected in the interstellar medium (ISM) increases, knowledge of their chemical properties is crucial in expanding our understanding of the chemistry of space. In this work we build on a previous study done in our group to examine the excited-state properties of five anions likely to exist in the ISM: SiCCN(-), CSiCN(-), CCSiN(-), SiCN(-), and SiNC(-). Our coupled cluster results indicate that SiCCN(-) and SiNC(-) possess dipole-bound singlet excited states while SiCCN(-) also has one valence state and CCSiN(-) potentially has two. Nearly all of the associated transition energies fall within the visible to near-IR region of the electromagnetic spectrum, making them applicable to the study of phenomena such as the diffuse interstellar bands.     

Test applications of a new SCF method for excited states

In order to test a recently proposed technique for deriving orthogonality-constrained HF wave functions for excited states, several applications to molecular systems, have been made and the results compared with those provided by other SCF techniques.     

On the solution of the quantum mechanical two-electron problem by direct calculation of the natural orbitals

The wave function of the ground state of the H₂ molecule is calculated directly in its natural expansion form, the approximate natural orbitals (NO) being expressed as linear combinations of Slater type functions centered at the midpoint of the molecule. One obtains as total energy (in the equilibrium distance) – 1.168 a.u. (exact – 1.174 a.u.) which seems to be the best one-center result for H₂ known so far; 96% of the binding energy is accounted for. The accuracy of this approach is limited due to the rather slow convergency of the onecenter expansion of the orbitals. The Hartree-Fock energy calculated with the same basis (–1.129 a.u.) is about as much in error with respect to the exact HF energy (– 1.136 a.u.) as the energy of the NO expansion with respect to the experimental one.

---

Zusammenfassung Die Wellenfunktion für den Grundzustand des H₂-Moleküls wird direkt in der Form ihrer natürlichen Entwicklung berechnet, wobei die natürlichen Orbitale (NO) angenähert werden als Linearkombinationen von Slater-Funktionen, die um den Schwerpunkt des Moleküls definiert sind. Die Energie des Zustands im Gleichgewichtsabstand ergibt sich zu – 1,168 a. u. (exakt – 1,174 a. u.), was anscheinend der beste bisher bekannte Wert für eine Einzentrenentwicklung des H₂ ist; 96% der Bindungsenergie werden erfaßt. Die Genauigkeit der Rechnung wird eingeschränkt durch die langsame Konvergenz der Einzentrenentwicklung der Orbitale. Die Hartree-Fock-Energie, berechnet mit der gleichen Basis (– 1,129 a. u.), unterscheidet sich von der exakten Hartree-Fock-Energie (– 1,136 a. u.) um etwa den gleichen Betrag wie diejenige der NO-Entwicklung von der experimentellen Energie.

---

Résumé La fonction d'onde de l'état fondamental de la molecule H₂ est calculée directement dans son developpement naturel, les orbitales naturelles (NO) approchées étant représentées comme combinaisons linéaires des fonctions de Slater définies par rapport au centre de la molecule. On obtient – 1,168 a.u. pour l'energie totale à la distance de l'equilibre (la valeur exacte vaut – 1,174 a.u.) et on tient compte de 96% de l'energie de liaison. Cette valeur est probablement la meilleure obtenue jusqu'ici dans le cadre d'un developpement monocentrique. La precision de ce calcul est limitée due à la convergence lente du developpement monocentrique. L'écart de l'energie Hartree-Fock calculée dans la même base (– 1,129 a.u.) par rapport à la valeur exacte (– 1,136 a.u.) vaut à 0,001 a.u. près celui de l'energie du developpement naturel par rapport à l'energie experimentale.

     

SCF wavefunctions and energies for valence states and excited states of carbon and nitrogen

Wavefunctions and energies are reported for the (2s)ⁿ(2p)^m states of neutral carbon and nitrogen using the fixed double-zeta bases employed by Clementi to describe the ground state of these atoms. In addition, the wavefunctions and energies for a number of valence states are given, including the so-called sp³ valence state of carbon. Calculated energies of the valence states agree well with those obtained from experiment. The corresponding valence-state orbitals are useful in semi-empirical quatum-mechanical calculators, such as the maximum overlap method.     

Dipole transitions from excited S states of two-electron systems in time-dependent Hartree-Fock theory

Surface enhanced Raman scattering has been observed for O₂ molecules adsorbed on polydiacetylene single crystals. The spectral dependence of the Raman cross section suggests that the enhancement is due to a well-defined electronic transition of energy 2.39 eV which involves a significant degree of charge transfer from the conjugated polydiacetylene backbone to the O₂ molecule.     

Non-empirical SCF and Cl Study of the ground and excited states of thioformaldehyde

Ab initio SCF and Cl calculations are reported for ground and various low-lying Rydberg and valence excited states of thioformaldehyde H₂CS. A double-zeta basis of near Hartree-Fock quality is employed in this work and the importance of polarization functions is also assessed. The calculations indicate uniformly larger CX bond lengths in this system than for H₂CO in the corresponding electronic states; they also lind potential minima for H₂CS non-planar nuclear conformations in the (n,π^*) and (π,π^*) excited states but in each case the calculated inversion barriers are seen to be smaller than those encountered in formaldehyde. The vertical transition energies to the various excited states studied are also found to be significantly smaller in H₂CS than in H₂CO but the order of electronic states is concluded to be virtually identical for the two systems. The lowest-lying excited states are the ^3,1(n,π^*) species calculated at 1.84 and 2.17 eV respectively; the first two allowed transitions are indicated to be the Rydberg species (n,s_R) and (n,px_R) at 5.83 and 6.62 eV. These are followed by the two allowed transitions σ → π^* and π → π^* at 7.51 and 7.92 eV respectively, both well below the first ionization limit in H₂CS. The much smaller splitting between the ^3,1(π,π^*) species in H₂CS than in H₂CO is attributed to the relatively diffuse charge distribution of the sulfur atom compared to that of oxygen.     

Orbital relaxation in the Rydberg series of the lithium atom. An excited state SCF calculation

Self-consistent-field (SCF ) calculations for a series of Rydberg states (1s²ns)²S of the Li atom are performed using the generalized Brillouin theorem (GBT) method. The calculated energy is a proper upper bound to the excited state energy. The SCF term values of the Rydberg states are almost the same as those of the frozen-core approximation ones. The orbital behavior shows that the core is slightly expanded by the penetration of the Rydberg orbitals, and the higher Rydberg orbitals can be very well represented by the modified hydrogen-like orbitals.     

On the direct calculation of the time evolution of excited molecular states in the presence of nonadiabatic interactions

The possibility is explored of calculating the time evolution of a given initial molecular state, in the presence of sufficiently strong nonadiabatic interactions, with a fully quantum-mechanical approach. Two methods are presented. The first one is based on the determination of the molecular eigenstates, with expansion of the nuclear wavefunctions on a Hermite basis. The second method is based on the Padé 1,1 approximation of the time evolution operator and on a finite difference representation of the time-dependent nuclear wavefunctions. Both methods are applied to simple models of a diatomic molecule.     

On the electronically excited states of uracil

Vertical excitation energies in uracil in the gas phase and in water solution are investigated by the equation-of-motion coupled-cluster and multireference configuration interaction methods. Basis set effects are found to be important for converged results. The analysis of electronic wave functions reveals that the lowest singlet states are predominantly of a singly excited character and are therefore well described by single-reference equation-of-motion methods augmented by a perturbative triples correction to account for dynamical correlation.Our best estimates for the vertical excitation energies for the lowest singlet n --> pi* and pi --> pi* are 5.0 +/- 0.1 eV and 5.3 +/- 0.1 eV, respectively. The solvent effects for these states are estimated to be +0.5 eV and +/- 0.1 eV, respectively. We attribute the difference between the computed vertical excitations and the maximum of the experimental absorption to strong vibronic interaction between the lowest A" and A' states leading to intensity borrowing by the forbidden transition.     

The first excited triplet states of the nucleotide bases calculated with different semiempirical SCF schemes

The π-electron distributions, spin densities, and energies of the first triplets of the nucleotide bases, uracil, thymine, cytosine, adenine, and guanine, were investigated in various semiempirical approximations. Results are presented for calculations using the semiempirical form of the closed-shell SCF configuration interaction method, of the different orbitals for different spins (unrestricted Hartree–Fock) treatment, with and without spin projection, and of the Roothaan's open-shell procedure.