The following is an attempt to describe the idea behind one semi-empirical model (the embedded atom method) that describes bonding interactions in metals.

Atoms are attracted to positions in the metal crystal in which they can optimize the electron density that they experience from their neighbours (not too much, nor too little). In one popular model, this energy is called the embedding energy. The embedding energy is a function of the electron density, so directional bonds are not needed in this model (in contrast to, say, semiconductors). The bonding simply arises from the tendency of an atom to find an electrostatic environment that it likes.

A lot of effort has gone into the development of bonding in metals, with the goal of calculating their properties either for practical applications or to test the accuracy of fundamental models.

In practice one does not use the concept of a bond at all. The focus is on understanding (a) how energy, structure and derived properties change when defects, deformations or impurities are introduced into the metal, or (b) how atoms move in the metal in different circumstances.

At school level, one learns that the lattice enthalpy (energy) is a measure of the energy released per atom when a metal crystal is assembled from gaseous atoms. It is interesting to ask how much energy is required to extract one atom from the centre of the crystal back to the gas phase. Instinctively one imagines that it is similar to the lattice enthalpy. This is true for an ionic lattice, but for a metal it is only about 35% of this value. That difference explains better than anything why bonds are not a useful concept for metals.