Why is the formation of double helix between non complementary sequences disfavoured? The textbooks starts with the phenomena that double helix is an expression of the rules of chemistry. 


  Hydrogen bonds are important in determining the formation of specific
   base pairs in the double helix. However in single staranded DNA, the hydrogen bond donors and acceptors are exposed to solution and can form hydrogen bonds with water molecules. 
  When two single strands come together, these hydrogen bonds with water are broken and new hydrogen bonds between the bases are formed. Because the number of hydrogen bonds broken is the same as the number formed, these hydrogen bonds do not contribute substantially to driving the overall process of double helix-formation. However, they contribute greately to the specificity of binding.


I see the point that some neucleotides which form stable links are the complimentary bases (purine and pyramidine). Some bases do not form stable links. I did search my question online, I am not able to get near anything that's directly related or comprehensible at my level. So following is what I do not understand. 


  Suppose two bases that cannot form Watson-Crick base pairs are brought together. Hydrogen bonds with water must be broken as the bases come into contact. Because the bases are not complementary in structure, not all of these bonds can be simultaneously replaced by hydrogen bonds between the bases. Thus, the formation of a double helix between noncomplementary sequences is disfavoured.


What's the specificity mentioned here? What does it mean by non complementary in structure? An example is appreciated.
If you take a look at the base pairings, as shown below, you can see that adenine and thymine form two hydrogen bonds when they are matched, while guanine and cytosine form three hydrogen bonds when they are matched. If you try to match an adenine with cytosine or guanine with thymine, the orientation of hydrogens and lone pairs will not line up. 

For example, guanine has two hydrogens and one lone pair available for hydrogen bonding, while cytosine has one hydrogen and two lone pairs available for hydrogen bonding. They fit! However, if you try to match adenine (with one hydrogen and one lone pair available for hydrogen bonding) with cytosine, the elements will not match or line up, resulting in an energetically less-favorable interaction.

Summary: A base on one strand of DNA requires another, specific base to complement the exposed hydrogens and lone pairs to form the most most energetically favorable interactions.