2. Methods

The ab initio MP4 and CCSD(T) calculations were done using the Gaussian 94 program package [23]. For all systems under study, we calculate properties of the lowest singlet state.

For the MP energy terms, we used

(1) E2 = E(MP2) - E(SCF)

(2) E3 = E(MP3) - E(MP2)

(3) E4 = E(MP4SDTQ) - E(MP3).

Then, the F4 energy is given by

$${\frac{{{E2}}^{3}\left (-{E2}+2\,{E3}-{ E4}\right )}{\left (-{E2}+{E3}\right )^{3}}}$$ (4)

and the [2/2] energy is

$${\frac{{{E2}}^{3}}{-{{E2}}^{2}+{E3}\,{E2}+{ E4}\,{E2}-{{E3}}^{2}}}$$ (5)

The Padé approximation defined in (5) is obtained converting the formal power series E0 + E1 z + E2 z² + E3 z³ + E4 z⁴ + ... with E0 = E(SCF) and E1 = 0 to a [2/2] Padé approximant followed by setting z = 1. The $\Pi$2 approximation is given by

$$\Pi {\frac{E2^2}{2}} {\frac{\displaystyle E2 -E3 + \sqrt {(E2- E3)^2 - 4 \,(E2\,E4- E3^2)} }{\displaystyle E2\,E4 - E3^2 }}$$ (6)

The above mentioned criterion for the usability of the MP series and the derived energies F4, [2/2], and $\Pi$2 is that the total variation of the latter three energies given by

$$\Delta \vert \vert \vert \Pi \vert \vert \Pi \vert$$ (7)

satisfies

$$\Delta$$ (8)

The results of the calculations presented in [8] are summarized in Figure 1. It is seen that the above criterion enforces that for the accepted cases the errors are indeed of the order of 5 mH. If one would not take the criterion (8) into account, the errors are much larger as shown in Figure 1.

Correlation energies are obtained either by regarding the inner electrons as an effective potential (frozen core option) or dealing with all electrons (full core option) ³.

In the single point calculations with different basis sets some basis sets were used which are not included in Gaussian 94 via the EMSL Gaussian Basis Set Order Form [24].

The potential surfaces of the diatomic molecules were calculated with the Gaussian 94 program [23]. A 6-311G basis was chosen since the single-point calculations show that this produces a rather satisfactory approximation of the energies on the one hand, and is expected to be applicable to much larger molecules, on the other hand. For the calculations of spectroscopic constants three different approaches were examined. Two fits of a narrow region around the minimum by polynomials of degree 3 or 5 were used to calculate the five spectroscopic constants $\omega_{e}^{}$,$\omega_{e}^{}$x_e, B_e,$\alpha_{e}^{}$, D_e, the equilibrium distance R_e, the force constant k_e and the minimum total electronic energy U_e. The third approach using a four parameter Morse potential as fit function yielded the two constants $\omega_{e}^{}$ and $\omega_{e}^{}$x_e.

The calculated spectroscopic constants arise from a second order approximation of the rovibrational energy as follows:

(9) E_tot(v, J) = E_vib(v) + E_rot(v, J)

with

$$\omega_{e}^{} {\textstyle\frac{1}{2}} \omega_{e}^{} {\textstyle\frac{1}{2}}$$ (10)

and

(11) E_rot(v, J) = B_vJ(J + 1) - D_vJ²(J + 1)²

where

$$\alpha_{e}^{} {\textstyle\frac{1}{2}}$$ (12)

and

$$\beta_{e}^{} {\textstyle\frac{1}{2}}$$ (13)

For the notation of the various spectroscopic constants we followed the monograph of Hurley [25, chapter 1.4]. The centrifugal distortion constant D_e should not be confused either with the dissociation energy $\ensuremath{\mathbf{D}_0}$ or the depth $\ensuremath{\mathbf{D}_e}$ of the potential at the minimum.

The Morse data were obtained by a least square fit of a Morse potential

$$U(R)=U_e + \ensuremath{\mathbf{D}_e}(1-\exp(-\beta (R-R_e)))^2$$ (14)

with fit parameters $U_e,\ensuremath{\mathbf{D}_e},R_e,\beta$ to a suitable part of the calculated potential surface that was chosen according to the above criterion (8). A Fortran 77 program which used the NAG [26] routine E04FDF was used in the fitting procedure. We follow the spectroscopic praxis and choose cm^-1 as unit for energies and frequencies in formulas and tables. The vibrational frequency $\omega_{e}^{}$ and the first anharmonicity constant were then calculated from the parameters of the Morse potential using the formulas

$$\omega_{e}^{} \beta \sqrt{\frac{\hbar \ensuremath{\mathbf{D}_e}}{\pi c \mu}}$$ (15)

and

$$\omega_{e}^{} {\frac{\omega_e^2}{4\ensuremath{\mathbf{D}_e}}}$$ (16)

where $\mu$ is the reduced mass of the diatomic molecule.

The least square fitting of the polynomials to the potential curves was done using Maple [27],[28]. From the coefficients R_e,k_e,a,b of the fitted polynomial

$${\frac{k_e}{2}}$$ (17)

the spectroscopic constants were calculated in the following way [25, chapter 1.4]

$$\omega_{e}^{} {\frac{1}{2 \pi c}} \sqrt{\frac{k_e}{\mu}}$$ (18)

$${\frac{\hbar}{4 \pi c \mu R_e^2}}$$ (19)

$$\alpha_{e}^{} {\frac{a(B_e R_e)^3}{\omega_e^3}} {\frac{B_e^2}{\omega_e}}$$ (20)

$$\omega_{e}^{} {\frac{30 B_e^3 R_e^6 a^2}{\omega_e^4}} {\frac{6 B_e^2 R_e^4 b}{\omega_e^2}}$$ (21)

$${\frac{4 B_e^3}{\omega_e^2}}$$ (22)

In the case of a polynom fit of third degree we set in Eq.(17) b = 0 and c = 0.