13.4 The Nuclear-electronic Orbital (NEO) Approach: Integrating Electronic and Nuclear Quantum Effects in Quantum Chemistry

13.4.1 NEO-Hartree-Fock

The simplest method within the NEO framework is the Hartree-Fock (NEO-HF) method, where the total nuclear-electronic wavefunction is approximated as a product of electronic (𝚽0e(𝐱e)) and nuclear (𝚽0p(𝐱p)) Slater determinants composed of electronic and protonic spin orbitals, respectively:

𝚿NEO-HF(𝐱e,𝐱p)=𝚽0e(𝐱e)𝚽0p(𝐱p). (13.33)

In Eq. (13.33), 𝐱e and 𝐱p are collective spatial and spin coordinates of the quantum electrons and protons. The NEO-HF energy for a restricted Hartree-Fock (RHF) treatment of the electrons and a high-spin open-shell treatment of the quantum protons is

ENEO-HF= 2iNe/2hiie+iNe/2jNe/2(2(ii|jj)-(ij|ij))
+INphIIp+12INpJNp((II|JJ)-(IJ|IJ))-2iNe/2INp(ii|II). (13.34)

The i,j, indices denote spatial occupied electronic orbital, and the I,J, indices correspond to spatial occupied protonic orbitals. In Eq. (13.4.1), hije and (ij|kl) are conventional electronic core Hamiltonian and two-electron integrals, respectively, and the corresponding terms for quantum protons are defined analogously. The last term in Eq. (13.4.1) is the Coulomb interaction between the electrons and the quantum protons. The spatial electronic and protonic orbitals (ψie(𝐫e) and ψIp(𝐫p)) are expanded as linear combinations of electronic or protonic Gaussian basis functions (ϕμe(𝐫e) and ϕμp(𝐫p)):

ψie(𝐫e)= μNebfCμieϕμe(𝐫e), (13.35)
ψIp(𝐫p)= μNpbfCμIpϕμp(𝐫p). (13.36)

The lower-case Greek letters without and with primes denote basis functions for electrons and protons, respectively, and Cμie and CμIp are electronic and protonic MO expansion coefficients, respectively.

Analogous to the conventional electronic Hartree-Fock method, the electronic and protonic coefficients are determined by variationally minimizing the energy in Eq. (13.4.1) via the self-consistent field (SCF) procedure. This procedure leads to a set of coupled electronic and protonic HF-Roothaan equations:

𝐅e𝐂e= 𝐒e𝐂e𝐄e, (13.37)
𝐅p𝐂p= 𝐒p𝐂p𝐄p, (13.38)

where 𝐒e and 𝐒p are electronic and protonic overlap matrices, respectively. The electronic and protonic Fock elements in Eqs. (13.37) and (13.38) are given by

Fμνe= hμνe+ρλPλρe((μν|ρλ)-12(μλ|ρν))-μνPνμp(μν|μν), (13.39)
Fμνp= hμνp+ρλPλρp((μν|ρλ)-(μλ|ρν))-μνPνμe(μν|μν). (13.40)

The electronic and protonic density matrix elements in Eqs. (13.39) and (13.40) are defined as

Pνμe= 2iNe/2CνieCμie*, (13.41)
Pνμp= INpCνIpCμIp*. (13.42)

The generalization to the unrestricted Hartree-Fock (NEO-UHF) treatment of electrons is accomplished by introducing separate spatial orbitals for α and β electron spins.

The analytic gradients of the NEO-HF energyWebb:2002 with respect to the classical nuclear coordinates (or coordinates of the centers of the quantum proton basis functions) can be derived and implemented. These gradients allow geometry optimizations within the NEO framework.