Terminology for deep-seated feelings? I'm interested in human emotion. Particularly, we have many words for different emotions, such as love, hate, etc. The issue arises that I have not found words for some of the emotions I've heard my friends describe and which I too have felt. These feelings do not appear to be accessible to everybody, because I have spoken with numerous people and only two of them so far have not met me with a blank stare or tried to fit my description to commonly described emotions like 'love' or 'anger'. Generally, the emotions are much more complex than the behaviours those words entail

For example, one of these feelings is the sensation of something missing, the sensation that there's more to life, and which embodies itself as a sense of vague, omnipresent dissatisfaction. Both I and a friend of mine have had this sensation and when we are happy, it is still there. If we think about it, it easily sobers the sensation of happiness, but at the same time, it is 'deeper', for lack of a better term, than standard emotion because it does not exclude other emotions.

Another such emotion, which does not seem to have a name is one I have experienced. It is a strong sensation of attachment to a girl who I used to date. We broke up, but remained friends, and I weathered through the resentment to keep the friendship because she thinks in a similar way to me. The sensation is very strong and brings to mind the idea that, regardless of what happens, I want her to be happy, whether we date again, remain friends, or become strangers. It is not an urge or compulsion to action, and like the sense of dissatisfaction, is omnipresent. It is also not linked to her presence or absence. Perhaps empathy or respect is the best word for it.

I'm curious about the terminology for sensations that exist only in relation to a mental concept and nothing tangible. Largely, it seems that language is inadequate to convey feelings of this kind, but I'm intensely interested in the terminology that may exist for these sensations. There are more of them, and they generally correspond to simpler emotions and concepts, but they have a very distinct and complex nature when compared to both the sensations and behaviours associated with those simple emotions.

Are there words to communicate these feelings without grossly simplifying and misrepresenting them?
Angst seems to fit your first example about as well as I could imagine any one word fitting. Angst is somewhat complex and ambiguous because it's used in many contexts with different connotations. It's roughly a combination of smoldering frustration and creeping anxiety as I understand it. It's more of a latent restlessness than an overt, clear emotion – somewhat more trait-like than most emotions, which are generally more transient states. Its use in existentialism particularly suits the global feeling of incompleteness you describe. IMO, it's a more cognitive or conative term than most emotion words too, in that it is sometimes thought to arise from unresolved cognitive dissonance regarding one's identity and meaning in life. Emotions are more often automatic reactions to present circumstances, generally speaking. 

As I understand this contrast, the usual emotions may occur on a somewhat different level of conscious experience. The ordinary emotional level fluctuates and cycles like the stimuli and routines that provoke them, somewhat like echoes of the stimuli. One may focus on them, or choose to let them pass, depending on the extent to which they hold one's attention, I suppose. The more cognitive side of experience is somewhat more stable. In some ways, people are built to resist changing their minds in response to every new stimulus. To be fair, most don't want to have a strong emotional reaction to every stimulus either, but the resistance is stronger with respect to beliefs and attitudes, or schemata in general. (I've described this resistance somewhat further in my answer to, 'Can attention be improved to reduce the frequency of mistakes?')

Schemata are more heavily involved in emotions about the self and life in general. Arguably, these emotions about subjects are attitudes, and thus subsumed partially within schemata as a broader class...but I haven't got a citation for that; it's just my understanding of how these concepts intertwine. (See also my answer to, 'Is there a precise definition of &quot;attitude&quot;?') IMO, it explains why feelings like angst, which often reflect discontented attitudes toward oneself or life at large, don't fluctuate as much and resist attempts to cheer up. The problems these attitudes indicate generally can't be resolved by just having a little fun or forgetting about them, as people often do to cope with simpler, more circumstantial, negative emotions. These problems tend to require attitude change to resolve, which surely is a deeper process. I see no problem with referring to this as "deeper" BTW; depth psychology and existential psychology both use this term extensively to differentiate higher-level (i.e., more complex or developmentally advanced) motivational content from the simpler stuff people are more used to discussing casually, atheoretically, or from folk psychological perspectives. 

As for your second example, companionate love from Sternberg's triangular theory, or storge from John Lee's / Hendrick &amp; Hendrick's theory of love styles would seem to fit equally and more or less synonymously. Again, it's more of an attitude about an enduring identity (hers, rather than yours, or that of life itself) than an emotional reaction to present, transient circumstances, so relative stability and independence from daily fluctuations in emotional states is to be expected. I would say empathy and respect are components of the emergent property of companionate love, especially the intermediate component of intimacy (one half of companionate love, according to Sternberg). Here's the diagram:

 by Lnesa, I think...

Feel free to comment or edit if I've missed your point by addressing your examples too specifically or narrowly. If you have other feelings that need classifying, I'd be happy to see more questions like this.