A variety of methods have been developed to incorporate nuclear quantum effects into reaction rate constant calculations. These methods include quantum scattering theory quantum wave packet dynamics, quasiclassical trajectory methods, path-integral based reactive-flux methods, instanton theory, quantum transition state theory, and variational transition state theory (VTST) with semiclassical multidimensional tunneling (MT) transmission coefficients. While these methods differ in terms of accuracy, efficiency, and scope of applicability, we will compare the new method presented in this article to CVT/SCT, where CVT denotes the canonical variational theory form of VTST, and SCT denotes the small-curvature tunneling form of multidimensional tunneling, because this combination is a well-validated and widely applied method for practical calculations. In general, one distinguishes two kinds of TST, conventional TST (CTST) and VTST. Both forms of TST are dynamical theories that calculate the equilibrium flux through a dividing surface (the transition state) that separates reactants from products. The transition statebeing a hypersurface -is missing one degree of freedom. In CTST, the transition structure passes through the saddle point between reactants and products and is missing the saddle point's imaginary-frequency normal mode. In CVT, one defines a reaction path and a generalized free energy of activation as a function of location along the reaction path (the generalized free energy of activation is the free energy of activation calculated at any point on a reaction path, not just at a transition state), and the transition state is taken to pass through the reaction path at the point of maximum generalized free energy of activation and to be missing the degree of freedom correspond to the reaction path. The reaction coordinate was originally defined in Cartesian coordinates, but in most current work and the present work is taken as a curvilinear coordinate. A key consequence of the TST rate being an equilibrium flux is that it can be calculated from partition functions (PFs). In the present article we make another distinction, namely the effective potential used in the PFs. In CTST and CVT, the potential energy surface is the usual Born-Oppenheimer potential energy surface. Here we define a new kind of TST, denoted CNEO-TST, which takes the effective potential energy surface to be the surface that is obtained by using nuclear-electronic orbital ("NEO") theory with a constraint ("C") imposed on the expectation value of the quantum nuclear position. NEO treats not only the electrons but also key nuclei (here taken as the hydrogenic nuclei) quantum mechanically, and CNEO constrains the expectation values of the quantal nuclei in order to obtain a coordinate-dependent energy that can serve as an effective potential energy surface. The resulting effective potential energy surface is denoted as the constrained minimized energy surface (CMES) or as the CNEO effective potential energy surface and is interpreted as an addition of the quantum delocalization effect, which includes the ZPE, to the Born-Oppenheimer potential energy surface. Direct dynamics, which is "the calculation of rates or other dynamical observables directly from electronic structure information, without the intermediacy of fitting the electronic energies in the form of a potential energy function," is the most convenient method for rate constant calculations. High-level electronic structure methods such as coupled cluster theory with high-order excitations and an extended basis set can often give accurate potential energy surfaces (PESs), but their use for direct dynamics is usually prohibitively expensive.