Triatomic molecule

Triatomic Molecules.—Haak and Stuhl have examined the formation of NH(A n) and ND(A IT) in the 193 nm photodissociation of NH, and ND,. A study of the intensity of emission and of the rotationally resolved fluorescence spectra of the photofragments indicated that the dissociation involves a two-photon resonance. The NH(A Il) and ND(A ll) were produced rotationally hot, leaving little translational energy to be carried by the H(D) atoms. 

Shapiro, and A. Yogev, Chem. Phys. Lett., 1984, 107, 603. 

Donovan, C. Fotakis, M. Martin, C. B. McKendrick, and A. Hopkirk, Ouanium Electron. Eleclro-opi., Proc. Natl. Quantum Electron. Conf., 5th, 1981 (publ. 1983), 135. 

The photodissociation of O3 by both u.v. and visible radiation has been investigated using the techniques of CARS applied to the O2 fragment. Photolysis at 266 nm was shown to produce 02( Ag) with an anomalous propensity for even J rotational states, an effect discussed in terms of sym- 

Tabche-Fouhaile, M.-J. Hubin-Franskin, J. P. Delwiche, H. Irohlkh, K. Ito, P.-M. Guyon, and I. Nenner, X Chem. Phys., 1983, 79, 5894. 

Diatomic Molecules. For triatomic molecules there are terms involving Coriolis interactions between vibrational modes in the correction from Bg to Bz, and the theory of the preceding sections is required. For linear or for bent symmetrical XY, molecules the rotational constants of a single isotopic species are sufficient to define the molecular structure completely, and thus for such molecules spectroscopic methods are superior in precision to the electron diffraction method. Electron diffraction studies serve as a test not only of the electron diffraction technique but also of the comparability between and r%. Some examples are given in Table 4. Besides the molecules CS., CIO, and CO, for which details are given, a 

The agreement between electron difiraction and spectroscopic results in these two tables is in general satisfactory. In most cases the electron difiraction measurement is within one or two standard deviations of the very accurate spectroscopic result. Discrepancies of up to two or three parts per thousand between independent electron difiraction investigations, as are seen for Cl. and SO. in the tables, may well be due to small errors in the measurement of electron wavelength or camera height, which together determine the experimental scale factor of the difiraction pattern (cf. Chapter 1, Section 2). These results emphasize the difficulty of measuring the scale factor to, say, 0.1 % even in the most careful studies. 

Some Other Small Polyatomic Molecules. For a few other small polyatomic molecules, spectroscopy can also provide a more accurate zero-point average structure than can electron difiraction. Examples are NH., ND., 

NF BFa, CH4, and CD4. Of these all but NFj have been the subject of a detailed comparison between electron diffraction and spectroscopy. - Again the comparison is satisfactory, and confirms both the accuracy obtainable with care in the electron diffraction technique and the theory for interconversion of the interatomic distance parameters. 

Combination of Rotationai Constants with Elecfrcm Diffraction Data.— 

Covalent Bonding III Triatomic Molecules Bond Angles 

Triatomic molecules may be linear or bent (i.e. V-shaped or angular). The shape adopted by any particular molecule is that which is consistent with the minimization of its total energy. 

The three normal modes of linear X3- and YXY- (Dooh) type molecules 

The structure of the dimeric species (MX2)2 is known to be cyclic-planar  

Stretching band, whereas c/s-dioxo groups show two such bands. For example, cis-VO groups show one band at 907-876 (antisymmetric) and the other at 922-910 cm (symmetric). Similar results are reported Tor cis-MoOr (Ref. 177) and cis- WOT (Ref. 178) groups. The UOf, NpO , PuOf, and AmOj groups are linear and exhibit only one band in the 940-850 cm  

The relationship between the UO2 frequency and the U=0 distance has been studied by several other investigators. 

Metal dioxides produced by the sputtering technique in inert gas matrices take linear or bent O-M-O structures. Metal dinitrides produced by the same technique also take linear N-M-N (M = U and Fu ) structures although Th(N2) is triangular These structures are markedly different from 

Diatomic molecules are necessarily linear, but a triatomic molecule can be either linear like CO2 and HCN or bent like SO2 and H2O. Mulliken s correlation diagram procedure was extended to tri- and tetraatomic molecules by Walsh [1], who promulgated a set of simple but remarkably viable [2, 3, 4] rules for predicting whether or not a molecule will remain linear, from the effect of the departure from linearity on the energy of its occupied molecular orbitals. 

As in diatomics (Section 4.3), the molecular orbitals in polyatomic molecules may be expressed as linear combinations of atomic orbitals (AOs) centered on the nuclei. A minimal basis set of AOs contains all of the AOs that are occupied in the separated atoms [1, 2]. In the bent ozone molecule, for example, the separated O atoms have the ground state configuration (ls) (2s) (2p). The minimal basis set for ground-state O3 therefore consists of the Is, 2s, and three 2p orbitals centered on each of the oxygen nuclei (Fig. 7.1). The orientations selected for the 2p AOs in the basis set are of course arbitrary (aside from the constraint that basis AOs centered on any nucleus must be linearly independ- 

The electronic states must transform as one of the irreducible representations r of the molecular point group, and so linear combinations of the AOs in Fig. 7.1 must be found which transform as these representations. Such symmetry-adapted linear combinations (SALCs) may be obtained using the projection operator technique [3]. Application of the projection operator 

By applying these projection operators to the Is AOs 1sa and lsc in Fig. 7.1, we obtain the unnormalized SALCs 

Neglecting overlap between the Is AOs, we then have the normalized SALCs 

Application of the projection operators P(a2) or P(bj) to any of the Is AOs yields a null result, and the linear combinations P(ai) lsB , b2) lsB , and P(b2) lsc all either vanish or reproduce the unnormalized SALCs (7.3). Hence, only three linearly independent SALCs are generated from the three Is basis AOs, as expected. In an LCAO-MO-SCF calculation, the two SALCs of a symmetry will mix to yield the lowest two MOs of a symmetry, lla ) and 2ai (Fig. 7.2). 

The same principles that we have used for the description of diatomic molecules will now be used in a description of the electronic structures of triatomic molecules. However, let it be clear that in using molecular-orbital theory with any hope of success, we first have to know the molecular geometry. Only in very rare cases is it possible from qualitative molecular-orbital considerations to predict the geometry of a given molecule. Usually this can only be discussed after a thorough calculation has been carried out. 

There are 16 valence electrons—6 electrons (2s)2 (2p)4 from each oxygen atom and 4 electrons (2s)2 (2p)2 from the carbon atom. The molecule is linear and has a center of symmetry. Thus, we characterize the orbitals according to the symmetry operations which are found in DOTh. We have for the carbon orbitals, 

For the oxygen orbitals it is practical to form first the possible linear combinations, and then find the transformation properties of these combinations  

We now combine the oxygen orbitals with the appropriate carbon orbitals to obtain the molecular orbitals for COz. Remember that only orbitals which transform alike can be combined together. We have, for example, the jtu orbitals =aupC(iru) + (i ip O - O] ( ttu). The 

The 16 valence electrons occupy the lowest orbitals in the ground state, giving the configuration (lag )2(1ctu )2 (2ag )2 (2ctu )2 (1 iru )4 (1 )4. 

This work has been extended to higher-molecular-weight polymers, (HF)3 and 

Matrix IR spectra of (LiF) (where n = 2,3,4) were interpreted in terms of ring structures (rhombic D2/, Da , and D4/, for n = 2, 3, and 4, respectively) [265]. The Raman spectrum of (Nal)2 in the gasous state at 1084 K supports D2/, structure [266]. The matrix IR. spectra of (GeO)2 [267] and (KO)2 [268] suggest a rhomic structure (Daft) and a slightly out-of-plane rhombic structure (C2 ), respectively. 

The nitric oxide dimer, (NO)2, in Ar matrices exists as the cis form (1862 and 1768cm ) or the trans form (1740cm ), with the v(NO) shown in brackets [268a]. The IR spectrum of the cis dimer in the gaseous state has been assigned [268b]. 

IR spectra of cis and tmns isomers of (NO)2 ion produced in Ne matrices exhibit the v(NO) at 1619 and 1424 cm, respectively. The corresponding frequencies of the (N0)2 ion produced inNe matrices are at 1225 (cis) and 1227 cm trans) [268c]. In the gaseous phase, NO forms a weakly bound van der Waals dimer, and the intermolecular vibrations are observed at 429(5 ), 239(8 ), 134 (intermolecular stretch) and 117 cm (out-of-plane torsion) [268d]. 

Inert gases and hydrogen halides form very weakly interacting van der Waals complexes. As expected, the v(HF) of Ne—HF [269] and v(HCI) of Ne-HCl [270] are shifted only slightly (+0.4722 and+3024 cm , respectively) from their frequencies in the free state. Vibrational frequencies of analogous Ar complexes are also reported [271-275]. IR spectra of van der Waals complexes such as CO—CH4 [276], CO—OCS [277] and CO—BF3 [278] are available. 

---

Heller E J 1978 Photofragmentation of symmetric triatomic molecules Time dependent pictured. Chem. Phys. 68 3891... 

Feit M D and Fleck J A Jr 1983 Solution of the Schrddinger equation by a spectral method, energy levels of triatomic molecules J. Chem. Phys. 78 301-8... 

Bade Z and Light J C 1986 Highly exdted vibrational levels of floppy triatomic molecules—a discrete variable representation—distributed Gaussian-basis approach J. Chem. Phys. 85 4594... 

We employ the general scheme presented above as a starting point in our discussion of various approaches for handling the R-T effect in triatomic molecules. We And it reasonable to classify these approaches into three categories according to the level of sophistication at which various aspects of the problem are handled. We call them (1) minimal models (2) pragmatic models (3) benchmark treatments. The criterions for such a classification are given in Table I. 

The situation in singlet A electronic states of triatomic molecules with linear equilibrium geometry is presented in Figme 2. This vibronic structure can be interpreted in a completely analogous way as above for n species. Note that in A electronic states there is a single unique level for K =, but for each other K 0 series there are two levels with a unique character. 

Figure 3. Low-energy vibronic spectrum in a. 11 electronic state of a linear triatomic molecule, computed for various values of the Renner parameter e and spin-orbit constant Aso (in cm ). The spectrum shown in the center of figure (e = —0.17, A o = —37cm ) corresponds to the A TT state of NCN [28,29]. The zero on the energy scale represents the minimum of the potential energy surface. Solid lines A = 0 vibronic levels dashed lines K = levels dash-dotted lines K = 1 levels dotted lines = 3 levels. Spin-vibronic levels are denoted by the value of the corresponding quantum number P P = Af - - E note that E is in this case spin quantum number),...

Figure 5, Low-eriergy vibronic spectrum in a electronic state of a linear triatomic molecule. The parameter c determines the magnitude of splitting of adiabatic bending potential curves, is the spin-orbit coupling constant, which is assumed to be positive. The zero on the...

We find it convenient to reverse the historical ordering and to stait with (neatly) exact nonrelativistic vibration-rotation Hamiltonians for triatomic molecules. From the point of view of molecular spectroscopy, the optimal Hamiltonian is that which maximally decouples from each other vibrational and rotational motions (as well different vibrational modes from one another). It is obtained by employing a molecule-bound frame that takes over the rotations of the complete molecule as much as possible. Ideally, the only remaining motion observable in this system would be displacements of the nuclei with respect to one another, that is, molecular vibrations. It is well known, however, that such a program can be realized only approximately by introducing the Eckart conditions [38]. 

An alternative form of exact nonrelativistic vibration-rotation Hamiltonian for triatomic molecules (ABC) is that used by Handy, Carter (HC), and... 

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

T is a rotational angle, which determines the spatial orientation of the adiabatic electronic functions v / and )/ . In triatomic molecules, this orientation follows directly from symmetry considerations. So, for example, in a II state one of the elecbonic wave functions has its maximum in the molecular plane and the other one is perpendicular to it. If a treatment of the R-T effect is carried out employing the space-fixed coordinate system, the angle t appearing in Eqs. (53)... 

Thus the angle t plays the role analogous to that of the angle defining the orientation of the instantaneous molecular plane in triatomic molecules. Employing the relations (69) and (59) one obtains... 

In this case, the situation is essentially equivalent to that for triatomics molecules. (We shall always assume that Ur > uc the fommlas for the opposite case, Ur < uc, are obtained from those to be derived by interchanging simply... 

---