The “mechanics” of respiration, sometimes called exogenous breathing,  constitute “breathing,” the USE of the lungs and their related nerves muscles and bones (ribs etc) for moving oxygen, carbon dioxide, and other gases to and/or from the blood."

The way we use “breathing mechanics” ala deep breathing has reference to breathing patterns (holding, gasping, sighing), breathing rate, breathing depth (volume), locus of breathing (chest and diaphragm), breathing coordination (relationship of upper and lower breathing wave), breathing resistance (nose and mouth), and collateral muscle activity for breathing regulation (muscles other than the diaphragm).

“Breathing chemistry” has two main co-factors; Carbon dioxide and oxygen. Most know those functions so I will pass over them here.
1.   Respiratory chemistry related to carbon dioxide impact can be significantly monitored by measuring changes in exhaled carbon dioxide, as well as to track significant (but not all) aspects of breathing mechanics. This carbon dioxide balance (endtidal CO2 -5%) is influenced by biochemical terrain such as acid alkaline balance, or cell hydration. It is often is a major marker for CO2 influences as related to vascular dilation or constriction feeding specific parts of the human body including the brain and vital organs;High blood levels of CO2 causes the body to open the gates to let in more oxygen to the red blood cells/hemoglobin.  CO2 levels can be a major marker for appropriate respiratory chemistry in of itself but I believe it almost always has cofactors (see breathing mechanics below) that may be much more relevant depending on the breathing pattern dominance of lower abdominal or high chest dominant breathing.

2. Law of Mass Action as referred to by William Campbell Douglas is whereby the extra oxygen can be "packed" into the body tissues for later recovery and usage.  The CO2 levels referred to in hypocapnea or too little CO2 in the blood may not always be as relevant as our science text books have led us to believe and may more depend on breathing mechanics and nervous system balance ala parasympathetic-low abdominal or sympathetic - high chest breathing patterns.  So we have at least TWO aspects of deep breathing that can greatly influence our body. 

RELAXATION.
Though a major marker for therapeutic progress, breathing mechanics for relaxation is too often the exclusive focus of breathing training and learning. As noted in the Secrets of Optimal Natural Breathing it too often gets tied into professional or spiritual agenda.  Ease is more important then relaxation as ease facilitates relaxation but relaxation does not necessarily foster ease and balance.

This "relaxation" may lower stress response and save lives, be a shift in conscious awareness and cognition, or be no more then a kind of transient trance state. As a result, a large percentage of therapists and trainers who teach breathing can deregulate respiratory chemistry by inducing “overbreathing/unbalanced breathing" with their "relaxation" instructions to trainees, not realizing that they are inducing system-wide psycho-spiritual and psycho-physiological crisis through the creation or exacerbation of hypocapnia, i.e. carbon dioxide deficit. This often results in exacerbating respiratory faults, or distorting optimal breathing balance and coordination as well as confusing the client/ patient.

Unfortunately, based on this kind of "relaxation" dominant thinking, myths and misunderstandings about “good” breathing often constitute the “working knowledge” of health professionals and lay audiences alike.  

FACTS TO REMEMBER from Dr. Peter Litchfield in italics
The wrong combination of breathing rate and depth can lead to "overbreathing."  Overbreathing means losing too much carbon dioxide.  CO2 deficit means blood alkalosis, and alkalosis often means smooth muscle constriction. Smooth muscle constriction means blood vessel, bronchiole, coronary, gut constriction. 
Overbreathing can reduce mental and physical performance.
Overbreathing can trigger emotion and physical symptoms.

From Mike. Overbreathing/hyperventilation may sometimes be miss named and actually be what we call hyper inhalation which accounts for the Law Of Mass Action referred to previously.  

Good breathing means relaxation.  
No.  Good breathing is important in all circumstances, whether relaxed or not. 
Mike sees this "good" breathing as more parasympathetic dominant.

Learning good breathing requires relaxation.  
No.  This would mean that during most life circumstances, breathing is maladaptive. There still must often be a window of balanced breathing  within the ranges of life's circumstances and that often has the look and feel of being "centered" or "grounded" or both.

Diaphragmatic breathing is synonymous with good breathing.  
No.  In many instances one may begin to over-breathe as a result of switching from chest to diaphragm.
From Mike. Parasympathetic dominance is the key.

Good respiration is all about the mechanics of breathing.  
No.  Good breathing means ventilating in accordance with metabolic requirements.
From Mike. This good respiration may as well include nutrition, toxicity and emotions.

Diaphragmatic, deep, slow breathing means better distribution of oxygen. 
No.  Mechanics may look letter perfect, but oxygen distribution may be poor.

Underbreathing/shallow breathing, with the result of oxygen deficit or loss of personal power, is common.  
Yes but also to the contrary, overbreathing is common.

Good breathing translates into optimizing respiratory psychophysiology, and contrary to popular thinking, learning to breathe well does not simply mean deep, slow, diaphragmatic breathing in the context of learning how to relax.  Adaptive breathing means regulating blood chemistry, through proper ventilation of carbon dioxide, in accordance with metabolic and other physiologic requirements associated with all life activities and circumstances: relaxation or stress, rest or challenge, fatigue or excitement, attention or open-focus, playing or working.  Deregulated, unbalanced breathing chemistry, i.e., hypocapnia (CO2 deficiency) as a result of overbreathing, means serious physiological crisis involving system-wide compromises that involve physical and mental consequences of all kinds, to be examined later in this overview.  

Evaluating, establishing, maintaining, and promoting good respiratory chemistry is extremely important in any professional practice involving breathing training.  Good breathing chemistry establishes a system-wide context conducive to help optimize health and maximize performance.  The more you can measure the more you can manage.

Breathing training is invariably included as an important component of relaxation training, but does not in of itself constitute optimal relaxation or optimal respiration. Optimal breathing implies ALL factors are in place in any moment in time.

At least 8 factors must be addressed to ensure optimal deep breathing. Priority is dependant upon the condition and situation.

Each one influences the others.

One may be more necessary at any given moment depending upon several circumstances such as stress and or the task at hand.  Optimal breathing can present itself during times of stress and challenge even where relaxation is neither possible nor adaptive. 

Training for various tasks alters the ratios and needs for each of the 8 basic influences as it inputs a conditioning factor that changes mechanical and cellular needs in the moment.

Whether you call it optimal breathing, overbreathing, hyperventilation, unbalanced breathing, maladaptive breathing or sub-optimal breathing, many receive great breathing AND respiration improvement with our self help combination of Manual, DS, 176 Fundamentals Video.  They may need to work with us over the phone or come to our office and benefit from modern oxygen and carbon dioxide measuring technology plus in-person breathing development expertise.