What are hybrid orbitals

Orbitals of all types are simply mathematical functions that describe particular standing-wave patterns that can be plotted on a graph but have no physical reality of their own. Because of their wavelike nature, two or more orbitals (i.e., two or more functions /) can be combined both in-phase and out-of-phase to yield a pair of resultant orbitals which, to be useful, must have squares that describe actual electron distributions in the atom or molecule. 

d and f orbitals that you are familiar with are the most convenient ones for describing the electron distribution in isolated atoms because assignment of electrons to them according to the usual rules always yields an overall function /2 that predicts a spherically symmetric electron distribution, consistent with all physical evidence that atoms are in fact spherical. For atoms having more than one electron, however, the s,p,d, f basis set is only one of many possible ways of arriving at the same observed electron distribution. Because it happens to be the simplest description of the atom we use it, but it is not unique. 

In the case of a molecule such as BeF, we know from experimental evidence that the molecule is linear and therefore the electron density surrounding the central atom is no longer spherical, but must be concentrated along two directions 180° apart, and we need to construct a function t2 having these geometrical properties. There are any number of ways of doing this, but the simplest is to use a new set of functions v / (which we call hybrid orbitals) that are constructed by combining the atomic s,p,d, f functions that are already familiar to us. 

You should understand that hybridization is not a physical phenomenon it is merely a mathematical operation that allows us to describe the electron distribution about a bonded atom in terms of one particular set of functions that we prefer to use because it is convenient to do so. 

Linus Pauling (1901-1994) was the most famous American chemists of the 20th century and the author of the classic The nature of the chemical bond. His early work pioneered the application of X-ray diffraction to determine the structure of complex molecules he then went on to apply quantum theory to explain these observations and predict the bonding patterns and energies of new molecules. Pauling, who spent most of his career at Cal Tech, won the Nobel Prize for Chemistry in 1954 and the Peace Prize in 1962. 

---