Quartic equation, energy surfaces

Though this is a quartic equation, it is capable of explicit solution because of the absence of second and third degree terms. Trial-and-error enters, however, because (GSi)r and are mild functions of Tg and related Te, respectively, and aprehminary guess of Tg is necessaiy. An ambiguity can exist in interpretation of terms. If part of the enclosure surface consists of screen tubes over the chamber-gas exit to a convection section, radiative transfer to those tubes is included in the chamber energy balance, but convection is not, because it has no effect on chamber gas temperature. 

The calculation of third and fourth derivatives is attractive for two reasons these are the quantities needed in the lowest-order treatment of vibrational anharmonicities, and a quartic surface is the simplest to exhibit a double minimum, i.e. the simplest model of a reaction surface. Moccia (1970) did apparently first consider the SCF third derivative problem. A detailed derivation of SCF and MCSCF third derivatives was given by Pulay (1983a) independently, Simons and J )rgensen (1983) also considered the calculation of MCSCF third, and even fourth, derivatives in a short note. As pointed out in Section II, third derivatives of the energy require only the first derivatives of the coefficients, and are thus computationally attractive. By contrast, fourth derivatives require the solution of the second-order CP MCSCF equations. The only computer implementation so far is that of Gaw etal.( 984) for closed shells, although the detailed theory has been worked out for the MCSCF case... 

---