From Chemical Bonding to Confinement
due to Nonadiabatic Processes by theEnhancement of Quantum Fluctuations
Y. Takada
The physical nature of chemical bonding has beenstudied in the adiabatic approximation in which only
electronic motion contributes to the formation ofcovalent bonds. Although this approximation may
well apply to most realistic molecules and solids, amore comprehensive understanding of chemicalbonding will be obtained by investigating the roles ofquantum fluctuations of nuclei in the bond formation.
We have commenced such an investigation byconsidering a hydrogen-like molecule, (M+M+m-m-),composed of two unit-positive charge particles(‘protons’) with mass M and two electrons with mass
m. The mass ratio m/M is changed continuously fromzero (the adiabatic limit) to unity (the limit ofdipositronium), covering the issue of a biexciton or abound complex of two electrons and two holes in
between. We have employed the diffusion Monte
Carlo (DMC) simulations at zero temperature in orderto evaluate the exact values for various ground-state
properties such as the total energy E0 and the averageseparation of ‘protons’ (or the bond length), together
with the corresponding ones in the adiabatic
approximation.

Fig. 1. Ground-state energy of a hydrogen-like molecule(M+M+m-m-) in both DMC and the adiabatic approximation as afunction of the mass ratio m/M.

In Fig. 1, the calculated result for E0 in DMC isshown as a function of m/M by the solid curve. The
obtained E0 increases monotonically with increasingm/M, but it is always smaller than the upper limit of
the energy for the existence of a bound state of thisfour-body problem (the dotted curve), namely, E0
M/(m+M). The adiabatic approximation provides avery accurate value for E0 with relative errors lessthan only 1% as long as m/M is smaller than 0.1,
convincing us that the binding mechanism for thismolecule in this range of m/M is nothing but thechemical bonding in the usual argument [1].

The decrease of the binding energy withincreasing m/M is associated with the rapid increase of
the bond length as shown in Fig. 2. Thus the vibronsor the zero-point quantum fluctuations of ‘protons’ arefound to hamper this chemical bonding by weakening
the binding energy and widening the bond length.

Fig. 2. Average distance between two ‘protons’ Rp inuints of the Bohr radius as a function of m/M. The inset
shows the relative variation Rp/ Rp .

For m/M larger than about 0.2, the adiabatic
approximation becomes too crude to support any
bound states in the system. Concomitant with thisresult for E0, the variation of ‘proton’ positions is aslarge as the average separation of two ‘protons’. Thisimplies that proton motion has changed its characterwith the increase of m/M; at m/M<<1, ‘protons’ arelocalized and separated to each other by the bondlength, while for m/M larger than about 0.2, they
overlap in the whole molecule in just the same way aselectrons.
Since the adiabatic potential cannot afford anenough attractive energy to bind ‘protons’ form/M>0.2, the existence of a bound state (or theconfinement of this four-body system) as
demonstrated by DMC suggests that nonadiabatic
processes or the effects due to retardation of electron
response to ‘proton’ motion should play a crucial rolein the binding mechanism for m/M in this region [2].
Finally we note that the parameter m/M controls thetwo competing effects on the binding mechanism; oneis to weaken the binding force through enhancing
quantum fluctuations and the other is to strengthen it
through promoting nonadiabatic processes. The
crossover takes place at m/M around 0.2.
References[1] K. Ruedenberg, Rev. Mod. Phys. 34, 326 (1962).
[2] Y. Takada and T. Cui, J. Phys. Soc. Jpn. 72, 2671
(2003).
Authors
Y. Takada and T. CuiaaJilin University.