Antisymmetric states excitation

Although the previous discussion has focused on ground states, the DMC method can also be applied to the calculation of electronically excited states. This is most simply achieved using the fixed-node approximation. Note that the ground state of a fermion system is itself an excited state. It is the lowest antisymmetric state of the system. 

While the above process is of great scientific interest, practically speaking we usually want to separate a racemic mixture of the B and B enantiomers, that is, our typical initial state is a racemate. If we were to use the scenario of Section 8.2 to accomplish this separation one would have first to prepare the BAB adduct in a pure state. Since the preparation of fire BAB adduct, and especially its separation from the BAB and B AB adducts that would inevitably accompany its preparation, is not a trivial task, it is preferable to find control methods that could separate the B and B racemic mixture directly. In this section we outline a method that can achieve this much more ambitious task. The essential principles of this method remain the same as in Section 8.2, that is, excitation of a superposition of symmetric and antisymmetric states with respect to ah, the reflection operation. 

A number of interesting conclusions follow from Eq. (81). In the first place, we note that the superposition states decay at different rates, the symmetric state decays with an enhanced rate (F I T ), while the antisymmetric state decays at a reduced rate (r — 1)2). For F12 = T, the antisymmetric state does not decay at all. In this case the antisymmetric state can be regarded as a dark state in the sense that the state is decoupled from the environment. Second, we note from Eq. (81) that the state a) is coupled to the state j) through the splitting A, which plays a role here similar to the Rabi frequency of the coherent interaction between the symmetric and antisymmetric states. Consequently, an initial population in the state a) can be coherently transferred to the state j), which rapidly decays to the ground state. When A = 0, that is, the excited states are degenerate, the coherent interaction does not take place and then any initial population in a) will stay in this state for all times. In this case we can say that the population is trapped in the state u). 

We shall next consider whether or not the antisymmetric eigenfunction Hl for two hydrogen atoms (Equation 29b) would lead to an excited state of the hydrogen molecule. The perturbation energy is found to be... 

The V (OCO) ion has a structured electronic photodissociation spectrum, which allows us to measure its vibrational spectrum using vibrationally mediated photodissociation (VMP). This technique requires that the absorption spectrum (or, in our case, the photodissociation spectrum) of vibrationally excited molecules differ from that of vibrationally unexcited molecules. The photodissociation spectrum of V (OCO) has an extended progression in the V OCO stretch, indicating that the ground and excited electronic states have different equilibrium V "—OCO bond lengths. Thus, the OCO antisymmetric stretch frequency Vj should be different in the two states, and the... 

Figure 13. Photodissociation spectrum of V (OCO), with assignments. Insets and their assignments show the photodissociation spectrum of molecules excited with one quanmm of OCO antisymmetric stretch, v" at 2390.9 cm . These intensities have been multiplied by a factor of 2. The shifts show that Vj (excited state) lies 24 cm below v ( (ground state), and that there is a small amount of vibrational cross-anharmonicity. The box shows a hot band at 15,591 cm that is shifted by 210 cm from the origin peak and is assigned to the V" -OCO stretch in the ground state.

Following this argument, in the first- and second-excited states, the electrons are placed in the Is and 2s orbitals. The antisymmetric spatial wave function has the lower energy, so that the first-excited state Pi(l, 2) is a triplet state. 

Woodward and Hoffmann have first disclosed that the thermal (4M+2)-cyclization (and also the photochemical (4M)-cyclization) takes place via Type I process, and the photochemical (4m+2)-cyclization (and also the thermal (4m)-cyclization) via Type II process 51>. They called the former (Type I) process "disrotatory", while the latter (Type II) process was referred to as "conrotatory". They attributed this difference in selectivity to the symmetry of HO and SO MO in the ground-state and excited-state polyene molecules, respectively (Fig. 7.33). The former is symmetric with respect to the middle of the chain, and the latter antisymmetric, so that the intramolecular overlapping of the end regions having the same sign might lead to the Type I and Type II interactions, respectively. 

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